Chu-Vandermonde Identity/Examples/3 from 4 + 5/Proof 2
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Example of Use of Chu-Vandermonde Identity
- $\ds \binom 9 3 = \binom {4 + 5} 3 = \sum_{k \mathop = 0}^3 \binom 4 k \binom 5 {3 - k}$
Proof
| \(\ds \binom 9 3\) | \(=\) | \(\ds \binom {4 + 5} 3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {9!} {3! \times 6!}\) | Definition of Binomial Coefficient | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {9 \times 8 \times 7} {3 \times 2 \times 1}\) | Definition of Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds 84\) |
$\Box$
| \(\ds \sum_{k \mathop = 0}^3 \binom 4 k \binom 5 {3 - k}\) | \(=\) | \(\ds \binom 4 0 \binom 5 3 + \binom 4 1 \binom 5 2 + \binom 4 2 \binom 5 1 + \binom 4 3 \binom 5 0\) | Definition of Summation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {4!} {0! \times 4!} \times \dfrac {5!} {3! \times 2!} + \dfrac {4!} {1! \times 3!} \times \dfrac {5!} {2! \times 3!} + \dfrac {4!} {2! \times 2!} \times \dfrac {5!} {1! \times 4!} + \dfrac {4!} {3! \times 1!} \times \dfrac {5!} {0! \times 5!}\) | Definition of Binomial Coefficient | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \times \dfrac {5 \times 4} {2 \times 1} + \dfrac 4 1 \times \dfrac {5 \times 4} {2 \times 1} + \dfrac {4 \times 3} {2 \times 1} \times \dfrac 5 1 + \dfrac 4 1 \times 1\) | Definition of Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \times 10 + 4 \times 10 + 6 \times 5 + 4 \times 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 10 + 40 + 30 + 4\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 84\) |
$\blacksquare$