The number of positive integers k such that the constant term in the binomial expansion of 2 x 3 + 3 x k 12 , x ≠ 0 is 2 8 · l , where l is an odd integer, is ______.

Question

The number of positive integers k such that the constant term in the binomial expansion of 2 x 3 + 3 x k 12 , x &#8800; 0 is 2 8 &#183; l , where l is an odd integer, is ______.

Quantrex Academy · Accepted Answer

Correct answer: 0

For 2 x 3 + 3 x k 12 T r + 1 = C r 12 2 x 3 12 - r &#183; 3 x k r = C r 12 &#183; 2 12 - r &#183; 3 r &#183; x 36 - 3 r - k r For the term to be independent of x , 36 - 3 r - k r = 0 &#8658; r = 36 k + 3 Here k can be 0 , 1 , 3 , 6 , 9 for r to be a whole number. But for C r 12 &#183; 2 12 - r &#183; 3 r to be in the form of 2 8 &#183; l Only k = 3 , &#160; 6 satisfies the above relation.

The number of positive integers k such that the constant term in the binomial expansion of 2 x 3 + 3 x k 12 , x ≠ 0 is 2 8 · l , where l is an odd integer, is ______.

Correct answer

Step-by-step solution

More from Binomial Theorem

The number of positive integers k such that the constant term in the binomial expansion of 2 x 3 + 3 x k 12 , x &#8800; 0 is 2 8 &#183; l , where l is an odd integer, is ______.

Correct answer

Step-by-step solution

More from Binomial Theorem

The number of positive integers k such that the constant term in the binomial expansion of 2 x 3 + 3 x k 12 , x ≠ 0 is 2 8 · l , where l is an odd integer, is ______.