Question

Find the exponent of 7 in ${}^{100}{{c}_{50}}$.
A. 0
B. 1
C. 2
D. 3

Answer 1

Hint: We first find exponent of 7 in 100! We then find exponent of 7 in 50! Then we find exponent of 7 in ${}^{100}{{c}_{50}}$. We are only using 100! and 50! Because ${}^{100}{{c}_{50}}=\dfrac{100!}{\left( 100-50 \right)!50!}=\dfrac{100!}{50!50!}$ which contains 100! and 50!

Complete step by step solution: Before proceeding to the solution, we must remember a very basic formula from binomial theorem which is:
${}^{n}{{c}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$
We also know the formula to find the exponent of a prime in n! which is :
The exponent of a prime ‘p’ in n! is the largest integer k such that ${{p}^{k}}$divides n!
 The exponent of p in n! is given by
\[=\left[ \dfrac{n}{p} \right]+\left[ \dfrac{n}{{{p}^{2}}} \right]+\left[ \dfrac{n}{{{p}^{3}}} \right]+\ldots \ldots \ldots \]
According to the above formula,
 Exponent of 7 in 100! is :
\[=\left[ \dfrac{100}{7} \right]+\left[ \dfrac{100}{{{7}^{2}}} \right]+\left[ \dfrac{100}{{{7}^{3}}} \right]+\ldots \ldots \ldots \]
$=14+2+0+0+\ldots \ldots \ldots 0$
$=16$
{Where; x is greatest integer function less than or equal to x
Now we find the exponent of 7 in 50! which is
\[=\left[ \dfrac{50}{7} \right]+\left[ \dfrac{50}{{{7}^{2}}} \right]+\left[ \dfrac{50}{{{7}^{3}}} \right]+\ldots \ldots \ldots \]
$=7+1+0+0+\ldots \ldots \ldots $
$=8$
Exponent of 7 in ${}^{100}{{c}_{50}}=\dfrac{100!}{50!50!}=\dfrac{\text{Exp}\ \text{of}\ 7\ \text{in}\ 100!}{\left( \text{Exp}\ \text{of}\ 7\ \text{in}\ 50! \right)}$is
$=\dfrac{{{7}^{16}}}{{{7}^{8}}{{7}^{8}}}=\dfrac{{{7}^{16}}}{{{7}^{8+8}}}=\dfrac{{{7}^{16}}}{{{7}^{16}}}={{7}^{0}}=1$.
 Exponent of 7 in ${}^{100}{{c}_{50}}$ is 1.
Correct option (B).

Note: The exponent of p in n! is given by
\[=\left[ \dfrac{n}{p} \right]+\left[ \dfrac{n}{{{p}^{2}}} \right]+\left[ \dfrac{n}{{{p}^{3}}} \right]+\ldots \ldots \ldots \]
This is the direct formula we used to find exponent of a prime in n!
You must also remember one expansion which is very handy in some problems:
${{\left( x+y \right)}^{n}}={}^{n}{{c}_{0}}{{x}^{n}}{{y}^{0}}+{}^{n}{{c}_{1}}{{x}^{n-1}}{{y}^{1}}+{}^{n}{{c}_{2}}{{x}^{n-2}}{{y}^{2}}+\ldots \ldots \ldots +{}^{n}{{c}_{r}}{{x}^{n-r}}{{y}^{r}}+\ldots \ldots \ldots $
      general term
$\ldots \ldots \ldots +{}^{n}{{c}_{r}}{{x}^{0}}{{y}^{n}}$