I need to simplify the expression
$$\sum_{k = 1}^{10} k\binom{10}{k}\binom{20}{10 - k}$$
Thank you.
2026-04-03 02:30:05.1775183405
Binominal expression simplification
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$$\sum_{k=1}^{10}k\binom{10}{k}\binom{20}{10-k}=10\sum_{k=1}^{10}\binom{9}{k-1}\binom{20}{10-k}=10\sum_{k=0}^{9}\binom{9}{k}\binom{20}{9-k}=10\binom{29}{9}$$
The last equality is Vandermonde's identity.