Question

The number of terms that are integers in the binomial expansion of (√7 + ∛5 )³⁵ is
(A) 4
(B) 5
(C) 6
(D) 7

plz give proper explanation..

Answer 1

Answer:

           (C) 6

Step-by-step explanation:

By the Binomial Theorem

                  $\displaystyle\bigl(\sqrt7+\sqrt[3]5\bigr)^{35}=\sum_{k=0}^{35}\binom{35}k7^{k/2}5^{(35-k)/3}$

So there are 36 terms in total, from k=0 to k=35, but for a term to be an integer the exponents on the 7 and the 5 must be integers.

• Exponent of 7 an integer  ⇒  k/2 an integer  ⇒  k ≡ 0 (mod 2)
• Exponent of 5 an integer  ⇒  (35 - k)/3 an integer  ⇒  k ≡ 2 (mod 3)

These two facts together are equivalent to just

• k ≡ 2 (mod 6)

The values of k from 0 to 35 that satisfy this are

• 2, 8, 14, 20, 26, 32

As there are six of these value of k, there are 6 terms that are integers.