Binomial Theorem · Question 31

The least value of n for which the number of integral terms in the Binomial expansion of ( [3]7 + [12]11)n is 183, is :

1. 2184
2. 2172
3. 2196
4. 2148

Answer

(1) 2184

Solution

Tr+1 = nCr 7 n - r3 11 r12

Integral term r is a multiple of 12 and (n - r) is a multiple of 3.

r = 12m n - 12m divisible by 3 n must be a multiple of 3.

Then r = 0, 12, 24, all give integral terms.

Number of integral terms = [ n12 ] + 1 = 183

[ n12 ] = 182 2184 n 2195

Least multiple of 3 in this range: n = 2184

Answer: (1)