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Question:

Suppose det[ n k-0 Σ k n k-0 Σ ^ n C_ k k ^ 2; n k-0 Σ ^ n C_ k k n k-0 Σ ^ n C_ k 3^ 2 ]=0holds for some positive integer n.Then ∑_k-0^n ^nC_kK+1 equals.

6.20

9.00

4.30

2.00

Solution:

Correct option is A. 6.20
n(n+1)2 n2^n-1+n(n-1)2^n-2 n2^n-1 4^n =0
n(n+1)2-n^24-n^2(n-1)8=0
n=0or4(n+1)-2n-n(n-1)=0
4n+4-2n-n^2+n=0
3n-n^2+4=0
n^2-3n-4=0
(n-4)(n+1)=0
n=4
∑_r-0^4 ^4C_rr+1=∑_r-0^4^5C_r+15=2^5-15=315=6.20