The value of displaystyle∑r=0s displaystyle∑s=1n n Css Cr is

Question

The value of displaystyle∑r=0s displaystyle∑s=1n n Css Cr is (A) 3n-1 (B) 3n+1 (C) 3n (D) 3(3n-1)

Tardigrade Admin · Accepted Answer

undersetr le s displaystyle∑r=0s displaystyle∑s=1n n Css Cr = displaystyle∑s=1n n Cs( s C0+ s C1+ s C2+ ldots+ s Cs) = displaystyle∑s=1n n Cs 2s= displaystyle∑s=0n n Cs 2s- n C0 20 =(1+2)n-1=3n-1

Q. The value of $r = 0 \sum s s = 1 \sum n^{n} C_{s}^{s} C_{r}$ is

$3^{n} - 1$

$3^{n} + 1$

$3^{n}$

$3 (3^{n} - 1)$

$r \leq s r = 0 \sum s s = 1 \sum n^{n} C_{s}^{s} C_{r}$
$= s = 1 \sum n^{n} C_{s} (^{s} C_{0} +^{s} C_{1} +^{s} C_{2} + \dots +^{s} C_{s})$
$= s = 1 \sum n^{n} C_{s} 2^{s} = s = 0 \sum n^{n} C_{s} 2^{s} -^{n} C_{0} 2^{0}$
$= (1 + 2)^{n} - 1 = 3^{n} - 1$

Q. The value of r=0∑s​s=1∑n​nCs​sCr​ is

3n−1

3n+1

3n

3(3n−1)