Know the property?

Algebra Level 3

${n \choose 0 } - {n \choose 1 } + {n \choose 2 } - {n \choose 3 } + ........ + (-1)^{r}{n \choose r } = 28$

then n=?

1 solution

Pranjal Jain
Nov 6, 2014

Just guessed it by using the fact that $\sum_{r=0}^n (-1)^r\binom{n}{r}=0$

First I thought that maybe $\binom{n}{n-1}-\binom{n}{n}=28$ which gave n=29 but on substitution, I got it to be -28.

Then I tried $\binom{n}{n-2}-\binom{n}{n-1}+\binom{n}{n}=28$ which gave me n=9.

Luckily this time it was correct! I also found in this process that r=6