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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 (16.8k points)
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1 Answer

+1 vote
$\sum \binom{n}{r}$ 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.
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