The Gateway to Computer Science Excellence
+2 votes
33 views

The value of  $^{13}C_{3} + ^{13}C_{5} + ^{13}C_{7} +\dots + ^{13}C_{13}$  is

  1. $4096$
  2. $4083$
  3. $2^{13}-1$
  4. $2^{12}-1$
in Combinatory by Boss (17.5k points)
edited by | 33 views

1 Answer

+2 votes
$\sum \binom{n}{r}$ from (r=0 to r=n) is equal to 2^n.

it includes both even and and odd terms . ( sum of odd terms + sum of even terms = 2^n).......(1)

and also sum of even terms = sum of odd terms..........(2).

solving  eq.(1) and eq (2) we can say that sum of odd terms = 2^(n-1).

so $\sum_{r=1}^{13} \binom{13}{r}$ = 2^(13-1)= 4096.

but we need the sum from $\binom{13}{3} +\binom{13}{5} +.......\binom{13}{13}$

since 2^(13-1) includes terms from $ \binom{13}{1} + \binom{13}{3} +\binom{13}{5} +.......\binom{13}{13}$

. so just subtract $\binom{13}{1}$ from 4096.

==> 4096-13= 4083 ans.
by (143 points)
Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,737 questions
57,376 answers
198,519 comments
105,311 users