Question

Find the exponent of $10$ in ${}^{75}{C}_{25}$

Answer 1

${}^{75}{C}_{25}=\frac{75!}{25!(75-25)!}$

$=\frac{75!}{25!50!}$

Now we have to count powers of 10 in $75!,25!,50!$

$10=5\times 2$

Now number of powers of 5 will be deciding factor to obtain as exponent of 2 > Exponent of 5

Exponent of 5 in 75!

$=\frac{75}{5}+\frac{75}{5^2}+\frac{75}{5^3}+........$

$=15+3+\frac{75}{125}$

$=15+3+0$ [As x=0 when 0 $\le$ x < 1]

$=18$

Exponent of 5 in 50!

$=\frac{50}{5}+\frac{50}{25}+\frac{50}{125}+........$

$=10+2+0$

$=12$

Exponent of 5 in 25!

$=\frac{25}{5}+\frac{25}{25}+\frac{25}{125}+........$

$=5+1+0$

$=6$

Hence we have

$\frac{{10}^{18}}{{10}^{12}\times {10}^6}=\frac{{10}^{18}}{{10}^{12+6}}=\frac{{10}^{18}}{{10}^{18}}=1={10}^0$

Hence exponent of 10 will be 0 in ${}^{75}{C}_{25}$