The value of C 50 100 51 + C 51 100 52 + ? + C 100 100 101 is: [2026]

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Solution

Q.
The value of $\frac{{}^{100}C_{50}}{51} + \frac{{}^{100}C_{51}}{52} + \dots + \frac{{}^{100}C_{100}}{101}$ is: [2026]

1 $\frac{2^{100}}{101}$

2 $\frac{2^{100}}{100}$

3 $\frac{2^{101}}{101}$

4 $\frac{2^{101}}{100}$

Ans.
(1)

$S = \sum_{r = 50}^{100} \frac{{}^{100}{C_{r}}}{r + 1} = \sum_{r = 50}^{100} \frac{1}{r + 1} \cdot \frac{r + 1}{101} .^{101} C_{r + 1}$

$S = \frac{1}{101} \sum_{r = 50}^{100} {}^{101}C_{r + 1}$

$= \frac{1}{101} \times \frac{2^{101}}{2} = \frac{2^{100}}{101}$

$Option (1)$

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