probability of drawing a 5 card flush given n cards Given $n$ random cards from a standard $52$ card deck, what is the probability of getting at least a 5 card flush within those $n$ cards? $n$ would be 5 &lt;= $n$ &lt; 17. The probability would get closer and closer to 1 as $n$ approaches 17. Basically, I need an equation to compute the increasing probability of getting at least a 5 card flush as you draw more and more cards from a 52 card deck.

Is it simply $$\frac {(^4C_1*   ^{13}C_5)}{^{52}C_n}$$
I've got kind of a dumb answer. I just wrote a program that counts the number $f$ of non-flush hands for a hand of $n$ cards and the total number of possible hands $t$ to arrive at the probability $P=1-f/t$.

! flush.f90 -- Calculates probability of getting a flush
module FlushFun
   use ISO_FORTRAN_ENV
   implicit none
   integer choose(0:13)
   contains
      function nfree(n)
         integer n
         integer(INT64) nfree
         integer n1, n2, n3
         integer(INT64) L1, L2, L3
         nfree = 0
         do n1 = max(0,n-12), min(4,n)
            L1 = choose(n1)
            do n2 = max(0,n-n1-8), min(4,n-n1)
               L2 = L1*choose(n2)
               do n3 = max(0,n-n1-n2-4), min(4,n-n1-n2)
                  L3 = L2*choose(n3)
                  nfree = nfree+L3*choose(n-n1-n2-n3)
               end do
            end do
         end do
      end function nfree
      function ntotal(n)
         integer n
         integer(INT64) ntotal
         integer i
         ntotal = 1
         do i = 1, N
            ntotal = ntotal*(52-i+1)/i
         end do
      end function ntotal
      subroutine init_params
         integer i
         integer(INT64) j
         integer(INT64) fact(0:13)
         fact = [(product([(j,j=1,i)]),i=0,13)]
         choose = [(fact(13)/fact(i)/fact(13-i),i=0,13)]
      end subroutine init_params
end module FlushFun

program flush
   use FlushFun
   implicit none
   integer(INT64) free, total
   real(REAL64) probability
   integer n
   call init_params
   write(*,'(a)') '\begin{array}{rrrl}'
   write(*,'(a)') '\text{Cards} &amp; \text{Non-Flush} &amp; \text{Total} &amp; \text{Probability}\\'
   write(*,'(a)') '\hline'
   do n = 4,17
      free = nfree(n)
      total = ntotal(n)
      probability = 1-real(free,REAL64)/total
      write(*,'(*(g0))') n, ' &amp; ', free,' &amp; ',total,' &amp; ',probability,' \\'
   end do
   write(*,'(a)') '\end{array}'
end program flush


$$\begin{array}{rrrl}
\text{Cards} &amp; \text{Non-Flush} &amp; \text{Total} &amp; \text{Probability}\\
\hline
4 &amp; 270725 &amp; 270725 &amp; 0.0000000000000000 \\
5 &amp; 2593812 &amp; 2598960 &amp; 0.19807923169267161E-002 \\
6 &amp; 20150884 &amp; 20358520 &amp; 0.10198973206303807E-001 \\
7 &amp; 129695332 &amp; 133784560 &amp; 0.30565769323455561E-001 \\
8 &amp; 700131510 &amp; 752538150 &amp; 0.69639844837102283E-001 \\
9 &amp; 3187627300 &amp; 3679075400 &amp; 0.13357924113216058 \\
10 &amp; 12234737086 &amp; 15820024220 &amp; 0.22662968679070705 \\
11 &amp; 39326862432 &amp; 60403728840 &amp; 0.34893320019744667 \\
12 &amp; 104364416156 &amp; 206379406870 &amp; 0.49430799449026441 \\
13 &amp; 222766089260 &amp; 635013559600 &amp; 0.64919475199817445 \\
14 &amp; 364941033600 &amp; 1768966344600 &amp; 0.79369814767023128 \\
15 &amp; 418161601000 &amp; 4481381406320 &amp; 0.90668912929163414 \\
16 &amp; 261351000625 &amp; 10363194502115 &amp; 0.97478084575449575 \\
17 &amp; 0 &amp; 21945588357420 &amp; 1.0000000000000000 \\
\end{array}$$
Not carefully checked, but at least it gives the right answers for $n\in\{4,5,17\}$. 

EDIT: To show how you could solve this problem by hand I wrote a program that really does find all the partitions of $n$ into $4$ integers in $[0,4]$. The most partitions you get is $8$ for $n=8$. Consider the partition $8=4+2+2+0$. In the table below this is represented as $4$ clubs, $2$ diamonds, $2$ hearts and no spades, but there are actually
$$\frac{4!}{1!2!1!}=12$$
Ways to choose the suits. For the given choice of suits, there are $\binom{13}{4}=715$ ways to select $4$ clubs, $\binom{13}{2}=78$ ways to select $2$ diamonds,  $\binom{13}{2}=78$ ways to select $2$ hearts, and $\binom{13}{0}=1$ way to select $0$ spades, so there are $12\times715\times78\times78\times1=52200720$ possible non-flush hands with the $4-2-2-0$ distribution. Here is the program that shows these calculations:

! flush2.f90 -- Calculates probability of getting a flush
module FlushFun
   use ISO_FORTRAN_ENV
   implicit none
   integer choose(0:13)
   contains
      function nfree(n)
         integer n
         integer(INT64) nfree
         integer n1, n2, n3, n4
         integer(INT64) L1, L2, L3, L4
         integer(INT64) term
         integer current, w1, w2, w3, w4
         nfree = 0
         write(*,'(a)') '$$\begin{array}{rrrr|r|rrrr|r}'
         write(*,'(a)') '\rlap{\text{Number}}&amp;&amp;&amp;&amp;'// &amp;
            '&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\'
         write(*,'(a)') '\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;'// &amp;
            '\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;'// &amp;
            '\heartsuit&amp;\spadesuit&amp;\text{Total}\\'
         write(*,'(a)') '\hline'
         do n1 = min(4,n), (n+3)/4, -1
            L1 = choose(n1)
            w1 = 1
            current = 1
            do n2 = min(n1,n-n1), (n-n1+2)/3, -1
               L2 = choose(n2)
               current = merge(current+1,1,n2 == n1)
               w2 = w1*current
               do n3 = min(n2,n-n1-n2), (n-n1-n2+1)/2, -1
                  L3 = choose(n3)
                  current = merge(current+1,1,n3 == n2)
                  w3 = w2*current
                  n4 = n-n1-n2-n3
                  L4 = choose(n4)
                  current = merge(current+1,1,n4 == n3)
                  w4 = w3*current
                  term = 24/w4*L1*L2*L3*L4
                  write(*,'(*(g0))') n1,'&amp;',n2,'&amp;',n3,'&amp;', &amp;
                     n4,'&amp;',24/w4,'&amp;',L1,'&amp;',L2,'&amp;',L3,'&amp;', &amp;
                     L4,'&amp;',term,'\\'
                  nfree = nfree+term
               end do
            end do
         end do
         write(*,'(*(g0))') '\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for ',n,' cards:}}&amp;',nfree
         write(*,'(a)') '\end{array}$$'
      end function nfree
      subroutine init_params
         integer i
         integer(INT64) j
         integer(INT64) fact(0:13)
         fact = [(product([(j,j=1,i)]),i=0,13)]
         choose = [(fact(13)/fact(i)/fact(13-i),i=0,13)]
      end subroutine init_params
end module FlushFun

program flush2
   use FlushFun
   implicit none
   integer(INT64) free
   integer n
   call init_params
   do n = 4,17
      free = nfree(n)
   end do
end program flush2


And here are the tables in prints out:
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;0&amp;0&amp;0&amp;4&amp;715&amp;1&amp;1&amp;1&amp;2860\\
3&amp;1&amp;0&amp;0&amp;12&amp;286&amp;13&amp;1&amp;1&amp;44616\\
2&amp;2&amp;0&amp;0&amp;6&amp;78&amp;78&amp;1&amp;1&amp;36504\\
2&amp;1&amp;1&amp;0&amp;12&amp;78&amp;13&amp;13&amp;1&amp;158184\\
1&amp;1&amp;1&amp;1&amp;1&amp;13&amp;13&amp;13&amp;13&amp;28561\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 4 cards:}}&amp;270725
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;1&amp;0&amp;0&amp;12&amp;715&amp;13&amp;1&amp;1&amp;111540\\
3&amp;2&amp;0&amp;0&amp;12&amp;286&amp;78&amp;1&amp;1&amp;267696\\
3&amp;1&amp;1&amp;0&amp;12&amp;286&amp;13&amp;13&amp;1&amp;580008\\
2&amp;2&amp;1&amp;0&amp;12&amp;78&amp;78&amp;13&amp;1&amp;949104\\
2&amp;1&amp;1&amp;1&amp;4&amp;78&amp;13&amp;13&amp;13&amp;685464\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 5 cards:}}&amp;2593812
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;2&amp;0&amp;0&amp;12&amp;715&amp;78&amp;1&amp;1&amp;669240\\
4&amp;1&amp;1&amp;0&amp;12&amp;715&amp;13&amp;13&amp;1&amp;1450020\\
3&amp;3&amp;0&amp;0&amp;6&amp;286&amp;286&amp;1&amp;1&amp;490776\\
3&amp;2&amp;1&amp;0&amp;24&amp;286&amp;78&amp;13&amp;1&amp;6960096\\
3&amp;1&amp;1&amp;1&amp;4&amp;286&amp;13&amp;13&amp;13&amp;2513368\\
2&amp;2&amp;2&amp;0&amp;4&amp;78&amp;78&amp;78&amp;1&amp;1898208\\
2&amp;2&amp;1&amp;1&amp;6&amp;78&amp;78&amp;13&amp;13&amp;6169176\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 6 cards:}}&amp;20150884
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;3&amp;0&amp;0&amp;12&amp;715&amp;286&amp;1&amp;1&amp;2453880\\
4&amp;2&amp;1&amp;0&amp;24&amp;715&amp;78&amp;13&amp;1&amp;17400240\\
4&amp;1&amp;1&amp;1&amp;4&amp;715&amp;13&amp;13&amp;13&amp;6283420\\
3&amp;3&amp;1&amp;0&amp;12&amp;286&amp;286&amp;13&amp;1&amp;12760176\\
3&amp;2&amp;2&amp;0&amp;12&amp;286&amp;78&amp;78&amp;1&amp;20880288\\
3&amp;2&amp;1&amp;1&amp;12&amp;286&amp;78&amp;13&amp;13&amp;45240624\\
2&amp;2&amp;2&amp;1&amp;4&amp;78&amp;78&amp;78&amp;13&amp;24676704\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 7 cards:}}&amp;129695332
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;0&amp;0&amp;6&amp;715&amp;715&amp;1&amp;1&amp;3067350\\
4&amp;3&amp;1&amp;0&amp;24&amp;715&amp;286&amp;13&amp;1&amp;63800880\\
4&amp;2&amp;2&amp;0&amp;12&amp;715&amp;78&amp;78&amp;1&amp;52200720\\
4&amp;2&amp;1&amp;1&amp;12&amp;715&amp;78&amp;13&amp;13&amp;113101560\\
3&amp;3&amp;2&amp;0&amp;12&amp;286&amp;286&amp;78&amp;1&amp;76561056\\
3&amp;3&amp;1&amp;1&amp;6&amp;286&amp;286&amp;13&amp;13&amp;82941144\\
3&amp;2&amp;2&amp;1&amp;12&amp;286&amp;78&amp;78&amp;13&amp;271443744\\
2&amp;2&amp;2&amp;2&amp;1&amp;78&amp;78&amp;78&amp;78&amp;37015056\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 8 cards:}}&amp;700131510
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;1&amp;0&amp;12&amp;715&amp;715&amp;13&amp;1&amp;79751100\\
4&amp;3&amp;2&amp;0&amp;24&amp;715&amp;286&amp;78&amp;1&amp;382805280\\
4&amp;3&amp;1&amp;1&amp;12&amp;715&amp;286&amp;13&amp;13&amp;414705720\\
4&amp;2&amp;2&amp;1&amp;12&amp;715&amp;78&amp;78&amp;13&amp;678609360\\
3&amp;3&amp;3&amp;0&amp;4&amp;286&amp;286&amp;286&amp;1&amp;93574624\\
3&amp;3&amp;2&amp;1&amp;12&amp;286&amp;286&amp;78&amp;13&amp;995293728\\
3&amp;2&amp;2&amp;2&amp;4&amp;286&amp;78&amp;78&amp;78&amp;542887488\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 9 cards:}}&amp;3187627300
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;2&amp;0&amp;12&amp;715&amp;715&amp;78&amp;1&amp;478506600\\
4&amp;4&amp;1&amp;1&amp;6&amp;715&amp;715&amp;13&amp;13&amp;518382150\\
4&amp;3&amp;3&amp;0&amp;12&amp;715&amp;286&amp;286&amp;1&amp;701809680\\
4&amp;3&amp;2&amp;1&amp;24&amp;715&amp;286&amp;78&amp;13&amp;4976468640\\
4&amp;2&amp;2&amp;2&amp;4&amp;715&amp;78&amp;78&amp;78&amp;1357218720\\
3&amp;3&amp;3&amp;1&amp;4&amp;286&amp;286&amp;286&amp;13&amp;1216470112\\
3&amp;3&amp;2&amp;2&amp;6&amp;286&amp;286&amp;78&amp;78&amp;2985881184\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 10 cards:}}&amp;12234737086
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;3&amp;0&amp;12&amp;715&amp;715&amp;286&amp;1&amp;1754524200\\
4&amp;4&amp;2&amp;1&amp;12&amp;715&amp;715&amp;78&amp;13&amp;6220585800\\
4&amp;3&amp;3&amp;1&amp;12&amp;715&amp;286&amp;286&amp;13&amp;9123525840\\
4&amp;3&amp;2&amp;2&amp;12&amp;715&amp;286&amp;78&amp;78&amp;14929405920\\
3&amp;3&amp;3&amp;2&amp;4&amp;286&amp;286&amp;286&amp;78&amp;7298820672\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 11 cards:}}&amp;39326862432
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;4&amp;0&amp;4&amp;715&amp;715&amp;715&amp;1&amp;1462103500\\
4&amp;4&amp;3&amp;1&amp;12&amp;715&amp;715&amp;286&amp;13&amp;22808814600\\
4&amp;4&amp;2&amp;2&amp;6&amp;715&amp;715&amp;78&amp;78&amp;18661757400\\
4&amp;3&amp;3&amp;2&amp;12&amp;715&amp;286&amp;286&amp;78&amp;54741155040\\
3&amp;3&amp;3&amp;3&amp;1&amp;286&amp;286&amp;286&amp;286&amp;6690585616\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 12 cards:}}&amp;104364416156
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;4&amp;1&amp;4&amp;715&amp;715&amp;715&amp;13&amp;19007345500\\
4&amp;4&amp;3&amp;2&amp;12&amp;715&amp;715&amp;286&amp;78&amp;136852887600\\
4&amp;3&amp;3&amp;3&amp;4&amp;715&amp;286&amp;286&amp;286&amp;66905856160\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 13 cards:}}&amp;222766089260
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;4&amp;2&amp;4&amp;715&amp;715&amp;715&amp;78&amp;114044073000\\
4&amp;4&amp;3&amp;3&amp;6&amp;715&amp;715&amp;286&amp;286&amp;250896960600\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 14 cards:}}&amp;364941033600
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;4&amp;3&amp;4&amp;715&amp;715&amp;715&amp;286&amp;418161601000\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 15 cards:}}&amp;418161601000
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
4&amp;4&amp;4&amp;4&amp;1&amp;715&amp;715&amp;715&amp;715&amp;261351000625\\
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 16 cards:}}&amp;261351000625
\end{array}$$
$$\begin{array}{rrrr|r|rrrr|r}
\rlap{\text{Number}}&amp;&amp;&amp;&amp;&amp;\rlap{\text{Hands in Suit}}&amp;&amp;&amp;&amp;\\
\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Ways}&amp;\clubsuit&amp;\diamondsuit&amp;\heartsuit&amp;\spadesuit&amp;\text{Total}\\
\hline
\hline&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\llap{\text{Hands for 17 cards:}}&amp;0
\end{array}$$
A simple approach!
You draw say 10 cards.
The probability that any of these cards is a particular suite is 1/4. Conversely the prob that any card in not say a diamond is 3/4.
Then what is probability that 5 cards are the same suite is inverse that any 5 cards are Not the same suite. This is simply 3/4 ^ 5 = 23.7%. So prob of a simple 5 card flush from 10 cards is  76.3% which to a card player  feels about right!
Cheers