Binomial Theorem
Weighted binomial coefficient sum
nta_pyq_2025_apr
Grade 12
Question:
If 1 2$\cdot$( 15 C 1$) + 2$2$\cdot$( 15 C 2$) + 3$2$\cdot$( 15 C 3 ) +$\ldots$+15 2$\cdot$( 15 C 15$) = 2$m$\cdot$3 n$\cdot$5 k , where m, n, k$\ in $N, then$m + n + k$is equal to :
Step-by-Step Solution
Key Concept: Use the general term of the binomial expansion for weighted binomial coefficient sum and impose the required coefficient or exponent condition.
15 2 15 15 14$\sumr$( Cr ) $\Rightarrow$ 15$\sum$r$Cr-1$(1)$r=1$$r=1$15 14 15$\sum$($r - 1 + 1)$$Cr-1$$r=1$15 15 13 14 15$\cdot$14$\sum$$Cr-2 + 15$$\sum$$Cr-1$$r=1$$r=1$13 14 15$\cdot$14$\cdot$$2 + 15$$\cdot$2 1 13 3$\cdot$2$(70 + 10)$1 13 3$\cdot$2$\cdot$80 1 1 17 3$\cdot$5$\cdot$2$m = 17n = 1$$k = 1$option (1)
Correct Answer: 1