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 ≠ 0 is 2 8 · l , where l is an odd integer, is ______.,

MARKS App · Accepted Answer

For 2 x 3 + 3 x k 12 T r + 1 = C r 12 2 x 3 12 - r · 3 x k r = C r 12 · 2 12 - r · 3 r · x 36 - 3 r - k r For the term to be independent of x , 36 - 3 r - k r = 0 ⇒ 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 · 2 12 - r · 3 r to be in the form of 2 8 · l Only k = 3 , 6 satisfies the above relation.

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