First slide

Permutations

Question

The exponent of 7 in the prime factorization of $^{100} C_{50}$ is

Moderate

A

0
B

2
C

4
D

none of these

Solution

We have $^{100} C_{50} = \frac{100!}{50! 50!}$
The exponent of $7$ in $50$ ! is $[\frac{50}{7}] + [\frac{50}{7^{2}}] = 7 + 1 = 8$
$[\frac{100}{7}] + [\frac{100}{7^{2}}] + [\frac{100}{7^{3}}]$ $= 14 + 2 = 16$
Thus, exponent of 7 in the prime factorization of $^{100} C_{50}$ is $16 - 2 (8) = 0$

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