The value of ∑ r = 0 7 7 C r r + 2 C 2 is equal to

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Question: Answered & Verified by Expert

The value of

\sum_{r = 0}^{7} \frac{^{7} C_{r}}{^{r + 2} C_{2}}

is equal to

MathematicsBinomial TheoremJEE Main

Options:

A $128$
B $\frac{251}{18}$
C $\frac{128}{3}$
D $\frac{11}{7}$

Solution:

2474 Upvotes Verified Answer

The correct answer is:

\frac{251}{18}

\sum_{r = 0}^{7} \frac{^{7} C_{r}}{\frac{(r + 2) (r + 1)}{2}}

= 2 \sum_{r = 0}^{7} \frac{^{7} C_{r}}{(r + 1) (r + 2)}

= 2 \sum_{r = 0}^{7} \frac{^{9} C_{r}}{8 \times 9}

= \frac{2}{8 \times 9} (2^{9} -^{9} C_{0} -^{9} C_{1})

= \frac{512 - 1 - 9}{36}

= \frac{502}{36}

= \frac{251}{18}

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