Question

Evaluate: \[ {}^{5}C_{0} + {}^{6}C_{1} + {}^{7}C_{2} + {}^{8}C_{3} + {}^{9}C_{4} + {}^{10}C_{5} + {}^{11}C_{6}. \]

Hint:

Remember the identity: $\sum_{k=0}^{m} {}^{k+r}C_{k} = {}^{m+r+1}C_{m}$. It is very useful for NAT combinatorics questions.

Answer 1

Correct Answer: 924

Step 1: Observe the pattern.
Each term follows the pattern: \[ {}^{n}C_{k} \quad \text{where } n = k+5. \]
Thus the sum becomes: \[ \sum_{k=0}^{6} {}^{k+5}C_{k}. \]
Step 2: Use known combinatorial identity.
Identity: \[ \sum_{k=0}^{m} {}^{k+r}C_{k} = {}^{m+r+1}C_{m}. \]
Here, \[ r=5, \quad m=6. \]
So, \[ \sum_{k=0}^{6} {}^{k+5}C_{k} = {}^{6+5+1}C_{6}. \]
\[ = {}^{12}C_{6}. \]
Step 3: Compute ${}^{12}C_{6}$.
\[ {}^{12}C_{6} = \frac{12!}{6!6!}. \]
\[ = 924. \]
Final Answer:
\[ \boxed{924}. \]