The value of ∑k=07 7k14k∑r=k14 rk14r, where nr denotes nCr, is

∑k=07 7Ck 14Ck∑r=k14 rCk⋅14Cr=∑k=07 7Ck∑r=k14 14−kCr−k=2141+127=67>76

The value of ∑k=07 7k14k∑r=k14 rk14r, where nr denotes nCr, is

∑k=07 7Ck 14Ck∑r=k14 rCk⋅14Cr=∑k=07 7Ck∑r=k14 14−kCr−k=2141+127=67>76

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The value of $\sum_{k = 0}^{7} [\frac{(\begin{array}{l} 7 \\ k \end{array})}{(\begin{array}{l} 14 \\ k \end{array})} \sum_{r = k}^{14} (\begin{array}{l} r \\ k \end{array}) (\begin{array}{l} 14 \\ r \end{array})]$ , where $(\begin{array}{l} n \\ r \end{array})$ denotes $^{n} C_{r}$ , is

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greater than $7^{6}$

$8^{7}$

greater than $7^{8}$

$6^{7}$

answer is A, B.

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Detailed Solution

$\begin{array}{l} \sum_{k = 0}^{7} (\frac{^{7} C_{k}}{^{14} C_{k}} \sum_{r = k}^{14}^{r} C_{k} \cdot^{14} C_{r}) = \sum_{k = 0}^{7} (^{7} C_{k} \sum_{r = k}^{14}^{14 - k} C_{r - k}) \\ = 2^{14} {(1 + \frac{1}{2})}^{7} = 6^{7} > 7^{6} \end{array}$

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The value of ∑k=07 7k14k∑r=k14 rk14r, where nr denotes nCr, is

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The value of $\sum_{k = 0}^{7} [\frac{(\begin{array}{l} 7 \\ k \end{array})}{(\begin{array}{l} 14 \\ k \end{array})} \sum_{r = k}^{14} (\begin{array}{l} r \\ k \end{array}) (\begin{array}{l} 14 \\ r \end{array})]$ , where $(\begin{array}{l} n \\ r \end{array})$ denotes $^{n} C_{r}$ , is