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Q. The exponent of $7$ in $^{100}C_{50}$ is

NTA AbhyasNTA Abhyas 2020Permutations and Combinations

Solution:

We know that, $^{100}C_{50}=\frac{100 !}{50 ! 50 !}$
The exponent of $7$ in $50!$ is
$\left[\frac{50}{7}\right]+\left[\frac{50}{7^{2}}\right]=7+1=8$
And the exponent of $7$ in $100!$ is
$\left[\frac{100}{7}\right]+\left[\frac{100}{7^{2}}\right]=14+2=16$
Thus, the exponent of $7$ in $^{100}C_{50}$ is $16-2\times 8=0$