Question

The exponent of 7 in the coefficient of the greatest term in the expansion of $(1+x{)}^{200}$ is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

0

The greatest coefficient will be
${}^{200}{C}_{100}$
$=\frac{200!}{100!.100!}$
Now the exponent of 7 in 200! will be
$\left[\frac{200}{7}\right]+\left[\frac{200}{49}\right]$
$=\left[28.57\right]+\left[4.08\right]$
$=28+4$
$=32$
The exponent of 7 in 100! will be
$\left[\frac{100}{7}\right]+\left[\frac{100}{49}\right]$
$=\left[14.28\right]+\left[2.04\right]$
$=14+2$
$=16$
Hence the power of 7 in $(100!{)}^2$ will be
$16+16$
$=32$
Hence the powers of seven in the numerator and denominator will be equal.
Hence answer is $0$.