What is the intuition for permuting $n$ objects where $p$ are alike If we have $n$ objects in which $p$ are objects are alike and rest are all different, then the number of permutations is $\frac{n!}{p!}$. Is there some intuition on how this is correct? why do we have to divide by $p!$

If you have $n$ objects where $p$ are alike. If you treat them like they're all different then the number of permutations is $n!$. But since there are $p$ objects alike this won't do. This is because the $n!$ will count the same permutation multiple times because it treats the $p$ alike objects like they're different. So to avoid double counting we divide by the number of ways to arrange $p$ objects, hence $\frac{n!}{p!}$. 
