Question

If x(x + 1)(x + 2)........ (x + 23) = Σ a_nxⁿ , n ∈ [n-1, 24] the number of coefficients a_n that are multiples of 3 is ________.

Answer 1

In elements of set {0,1,2,........23}

8 elements are of the form 3λ

8 elements are of the form 3λ + 1

8 elements are of the form 3λ - 1

Coefficient of x²⁴ is 1 (not divisible by 3)

Coefficient of x²² is not multiple of 3

Coefficient of x²¹ is multiple of 3

Coefficient of x⁸ is not multiple of 3

Coefficient of x⁷ is multiple of 3

Now all coefficient of power of x less than 7 are multiple of 3

Coefficient of x²⁴ , Coefficient of x²² ,......... Coefficient of x⁸ are not multiple of 3

Number of coefficient which are multiple of 3 are 24 – 9 = 15