First slide

Combinations

Question

$\underset{0 \leq i \leq j \leq 10}{\sum \sum}^{10} C_{j}^{j} C_{i}$ is equal to

Moderate

A

3¹⁰
B

3¹⁰ – 1
C

2¹⁰
D

2¹⁰ – 1

Solution

$\underset{0 \leq i \leq j \leq 10}{\sum \sum}^{10} C_{j}^{j} C_{i}$
$=^{10} C_{0} \times^{0} C_{0} +^{10} C_{1} (^{1} C_{0} +^{1} C_{1}) +^{10} C_{2} (^{2} C_{0} +^{2} C_{1} +^{2} C_{2}) +^{10} C_{3} (^{3} C_{0} +^{3} C_{1} +^{3} C_{2} +^{3} C_{3}) + \dots +^{10} C_{10} (^{10} C_{0} +^{10} C_{1} + \dots +^{10} C_{10}) =^{10} C_{0} \cdot 2^{0} +^{10} C_{1} \cdot 2^{1} +^{10} C_{2} \cdot 2^{2} + \dots +^{10} C_{10} \cdot 2^{10} = (1 + 2)^{10} = 3^{10}$

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