Binomial Theorem
Binomial series summation
nta_pyq_2025_apr
Grade 12
Question:
The sum of the series 2 $\times$ 1 $\times$ 20$C_{4}$- 3 $\times$ 2 $\times$ 20$C_{5}$+ 4 $\times$ 3 $\times$ 20$C_{6}$- 5 $\times$ 4 $\times$ 20$C_{7}$+$\ldots$+ 18 $\times$ 17 $\times$ 20$C_{20}$, is equal to
Step-by-Step Solution
Key Concept: Use the general term of the binomial expansion for binomial series summation and impose the required coefficient or exponent condition.
$(1 - x)$$20 = 20$$C_{0}$- 20$C_{1}$$x + 20$$C_{2}$x 2$\ldots$.$.+ 20$$C_{20}$x 20 20 (34) 20 20$(1 - x)$$C_{0}$$C_{1}$20 20 20 2 = - +$C_{2}$-$C_{3}$x +$C_{4}$x$\ldots$2 2 x x x Diff twice and put$x = 1$$20 = 6$-$C_{1}$$(2) + A$$A = 40 - 6 = 34$
Correct Answer: 34