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  4. The value of displaystyle∑r=110 r ⋅ ( n Cr/ n Cr-1) is equal to

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Q. The value of $\displaystyle\sum_{r=1}^{10} r \cdot \frac{{ }^n C_r}{{ }^n C_{r-1}}$ is equal to

Binomial Theorem

Solution:

$\displaystyle\sum_{r=1}^{10} r \cdot \frac{{ }^n C_r}{{ }^n C_{r-1}}=\displaystyle\sum_{r=1}^{10}(n-r+1)$
$=(n+1) \times 10-\frac{10 \times 11}{2}=10 n-45$