The value of k = 0 6 C 3 51 - k is equal to

Question

The value of $\sum_{k = 0}^{6} {}^{51 - k}{C_{3}}$ is equal to

Smart Achievers · Accepted Answer

(2)

$\sum_{k = 0}^{6} {}^{51 - k}C_{3} = {}^{51}C_{3} + {}^{50}C_{3} + {}^{49}C_{3} + {}^{48}C_{3} + {}^{47}C_{3} + {}^{46}C_{3} + {}^{45}C_{3}$

As we know, ${}^{n}C_{r} + {}^{n}C_{r - 1} = {}^{n + 1}C_{r} \Rightarrow {}^{n}C_{r - 1} = {}^{n + 1}C_{r} - {}^{n}C_{r}$

So, ${}^{51}C_{3} = {}^{52}C_{4} - {}^{51}C_{4}, {}^{50}C_{3} = {}^{51}C_{4} - {}^{50}C_{4}$ and ${}^{45}C_{3} = {}^{46}C_{4} - {}^{45}C_{4}$

$\Rightarrow$ $\sum_{k = 0}^{6} {}^{51 - k}C_{3} = {}^{52}C_{4} - {}^{51}C_{4} + {}^{51}C_{4} - {}^{50}C_{4} + \dots + {}^{47}C_{4} - {}^{46}C_{4} + {}^{46}C_{4} - {}^{45}C_{4}$

$= {}^{52}C_{4} - {}^{45}C_{4}$

Solution

Q.
The value of $\sum_{k = 0}^{6} {}^{51 - k}{C_{3}}$ is equal to

1 ${}^{51}{C_{3}} - {}^{45}{C_{3}}$

2 ${}^{52}C_{4} - {}^{45}C_{4}$

3 ${}^{52}C_{3} - {}^{45}C_{3}$

4 ${}^{51}C_{4} - {}^{45}C_{4}$

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Solution

Q. The value of ∑k=06C351-k is equal to

1 C351-C345

2 C452-C445

3 C352-C345

4 C451-C445