How many positive integer solutions satisfy the condition $y_1+y_2+y_3+y_4 < 100$ In preparation for an upcoming test I have come across the following problem and I am looking for some help with it just in case a question of its kind comes up on a evaluation. Thanks!

How many positive integer solutions satisfy the condition:$$y_1+y_2+y_3+y_4 < 100$$


Lets say the question was:
$$y_1+y_2+y_3+y_4 = 4$$
There is only $1$ solution:
$$1+1+1+1 = 4$$
Now lets assume it was:
$$y_1+y_2+y_3+y_4 = 5$$
There are $4$ solutions:
$$1+1+1+2=5$$
$$1+1+2+1 = 5$$
$$1+2+1+1 = 5$$
$$2+1+1+1 = 5$$
Next assume it was:
$$y_1+y_2+y_3+y_4 = 5 $$
There are $10$ solutions:
$$1+1+1+3 = 6$$
$$1+1+2+2 = 6$$
$$1+1+3+1 = 6$$
$$1+2+2+1 = 6$$
$$1+2+1+2 = 6$$
$$1+3+1+1 = 6$$
$$2+1+1+2 = 6$$
$$2+1+2+1 = 6$$
$$2+2+1+1 = 6$$
$$3+1+1+1 = 6$$
There's a pattern:
$$n=4 \implies \binom{3}{3} = 1$$
$$n=5 \implies\binom{4}{3} = 4$$
$$n=6 \implies\binom{5}{3} = 10$$
$$n=7 \implies\binom{6}{3} = 20$$
$$n=8 \implies\binom{7}{3} = 35$$
Then by adding up all the terms up to $98$ you get the answer:
$$\sum_{i=3}^{98}\binom{i}{3} =\binom{99}{4}= 3764376$$
Indeed by Hockey-stick identity we have
$$\sum^n_{i=r}{i\choose r}={n+1\choose r+1} \qquad \text{ for } n,r\in\mathbb{N}, \quad n>r$$
