Question

Find the value of ${}^{15}{C}_2$+ 2 ${}^{15}{C}_3$ + 3 ${}^{15}{C}_4$ . . . . . 14 ${}^{15}{C}_{15}$

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Answer 1

Each term is of the term (r-1) ${}^{15}{C}_r$, where r varies from 2 to 15.

⇒ sum = $\sum _{r=2}^{15}(r-1)$ ${}^{15}{C}_r$

${}^n{C}_r$ = $\frac{n}{r}$${}^{n-1}{C}_{r-1}$

⇒ r${}^n{C}_r$ = n ${}^{n-1}{C}_{r-1}$

⇒ r${}^{15}{C}_r$ = 15 ${}^{14}{C}_{r-1}$

⇒ sum = $\sum _{r=2}^{15}r$ ${}^{15}{C}_r$ - $\sum _{r=2}^{15}$ ${}^{15}{C}_r$

= $\sum _{r=2}^{15}15$ ${}^{14}{C}_{r-1}$ - $\sum _{r=2}^{15}$ ${}^{15}{C}_r$

= 15(${}^{14}{C}_1$ + ${}^{14}{C}_2$..........${}^{14}{C}_{14}$) - (${}^{15}{C}_2$ + ${}^{15}{C}_3$..............${}^{15}{C}_{15}$)

= 15(${}^{14}{C}_0$ + ${}^{14}{C}_1$..............${}^{14}{C}_{14}$) - (${}^{14}{C}_0$)

- ((${}^{15}{C}_0$ + ${}^{15}{C}_1$+ .........${}^{15}{C}_{15}$) - (${}^{15}{C}_0$ + ${}^{15}{C}_1$))

= 15 $(2^{14}-1)-(2^{15}-16)$

= 15 $2^{14}$ - 2 $\times$ $2^{14}$ - 15 + 16

= 13 $\times$ $2^{14}$ + 1

= 212993