Let n Cr denote the binomial coefficient of xr in the expansion of (1+x)n. If displaystyle∑ k =010(22+3 k ) n C k =α .310+β .210, α, β ∈ R then α+β is equal to .

Question

Let n Cr denote the binomial coefficient of xr in the expansion of (1+x)n. If displaystyle∑ k =010(22+3 k ) n C k =α .310+β .210, α, β ∈ R then α+β is equal to . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Instead of n C k it must be 10 C k i.e. displaystyle∑ k =010(22+3 k ) 10 C k =α .310+β .210 LHS =4 displaystyle∑ k =010 10 C k +3 displaystyle∑ k =010 k ⋅ (10/ k ) ⋅ 9 C k -1 =4.210+3.10 .29 =19.210=α .310+β .210 ⇒ α=0, β=19 ⇒ α+β=19

Q. Let nCr​ denote the binomial coefficient of xr in the expansion of (1+x)n. If k=0∑10​(22+3k)nCk​=α.310+β.210,α,β∈R then α+β is equal to ______.

Instead of nCk​ it must be 10Ck​ i.e. k=0∑10​(22+3k)10Ck​=α.310+β.210 LHS=4k=0∑10​10Ck​+3k=0∑10​k⋅k10​⋅9Ck−1​ =4.210+3.10.29 =19.210=α.310+β.210 ⇒α=0,β=19⇒α+β=19

Q. Let $^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1 + x)^{n}$ . If $k = 0 \sum 10 (2^{2} + 3 k)^{n} C_{k} = α . 3^{10} + β . 2^{10}, α, β \in R$ then $α + β$ is equal to ______.