Question

Let \(\binom{n}{r}\) denote the number of ways of choosing \(r\) distinct objects out of \(n\) distinct objects. Then,

Hint:

The Hockey Stick Identity is very useful for sums of binomial coefficients: \[ \sum_{r=0}^{m}\binom{r+k}{k}=\binom{m+k+1}{k+1}. \]

Answer 1

Correct Answer: 462

Step 1: Spot the pattern.
The sum is $\binom50+\binom61+\binom72+\cdots+\binom{11}{6}$, where each term is $\binom{r+5}{r}$ for $r=0$ to $6$.

Step 2: Rewrite the terms.
Using $\binom{r+5}{r}=\binom{r+5}{5}$, the sum becomes $\sum_{r=0}^{6}\binom{r+5}{5}$.

Step 3: Hockey stick identity.
The rule $\sum_{r=0}^{m}\binom{r+k}{k}=\binom{m+k+1}{k+1}$ collapses the whole sum.

Step 4: Put $k=5$, $m=6$.
\[ S=\binom{6+5+1}{5+1}=\binom{12}{6} \]

Step 5: Compute.
\[ \binom{12}{6}=\frac{12!}{6!\,6!}=462 \]
\[ \boxed{462} \]