For calculating the value of choosing $r$ items from $n$ items where $q$ are of same kind, and we should take modulo $m$, where $m$ is a prime; I used the following relation:

$$\frac{\,^{n}P_{r}}{q!} \pmod{m}$$

For calculating this

  • I first calculated $n!$, then $n!\bmod{m}$

  • Next, I calculated $(n-r)!$ and multiplied it with $q!$, i.e $t = q!(n-r)!$

  • Then I multiplied the modular multiplicative inverse of $t$ with $n!$ and took the result modulo $m$ (i.e. $t^{-1}n!\bmod{m}$)

But I am not getting the correct answer.. E.g: if $n= 3$, $r = 2$, $q = 2$ then the expected result is $\frac{\,^{3}P_{2}}{2!}\equiv3\pmod{1000000007}$ but am getting $250000004$.. I can't understand my mistake here.. Thanks.


It's difficult to see what's going wrong here. How are you computing the multiplicative inverse of 2? Regardless of that, what value do you get for $2^{-1} \pmod{1000000007}$ ? (Since the modulus is odd, the answer should be immediately obvious.)


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