Solveeit Logo
Login

Question

Mathematics Question on permutations and combinations

C1Co+2C2C1+3C3C2+....+nCnCn−1\frac {C_1}{C_o} + 2 \frac {C_2}{C_1} +3\frac {C_3}{C_2} + .... +n \frac {C_n}{C_{n-1}} =

A

n(n−1)2\frac{n(n-1)}{2}

B

n(n+1)2\frac{n(n+1)}{2}

C

(n+1)(n+2)2\frac{(n+1)(n+2)}{2}

D

None of theseNone \,of \,these

Answer

n(n+1)2\frac{n(n+1)}{2}

Explanation

Solution

C1C0+2C2C1+3C3C2+…+n−CnCn−1\frac{C_{1}}{C_{0}}+2 \frac{C_{2}}{C_{1}}+3\frac{C_{3}}{C_{2}} +\ldots+ n-\frac{C_{n}}{C_{n-1}}
=nC1nC0+2nC2nC1+3nC3nC2+…+nnCnnCn−1=\frac{^{n}C_{1}}{^{n}C_{0}}+2 \frac{^{n}C_{2}}{^{n}C_{1}}+3\frac{^{n}C_{3}}{^{n}C_{2}}+\ldots+n \frac{^{n}C_{n}}{^{n}C_{n-1}}
=n1+2×n(n−1)2n+3×n(n−1)(n−2)3×2n(n−1)2+…+n×1n=\frac{n}{1}+2\times\frac{\frac{n\left(n-1\right)}{2}}{n}+3\times\frac{\frac{n\left(n-1\right)\left(n-2\right)}{3\times2}}{\frac{n\left(n-1\right)}{2}}+\ldots+n\times\frac{1}{n}
=n+(n−1)+(n−2)+…+1=n+\left(n-1\right)+\left(n-2\right)+\ldots+1
=∑n=n(n+1)2=\sum n=\frac{n \left(n+1\right)}{2}