Question

Evaluate the following:

\[\sum^5_{r = 1} {}^5 C_r\]

 

Answer 1

We have,

\[\sum^5_{r = 1} {}^5 C_r =^5 C_1 +^5 C_2 +^5 C_3 +^5 C_4 +^5 C_5\]

\[\Rightarrow \sum^5_{r = 1} {}^5 C_r =^5 C_1 +^5 C_3 +^5 C_3 +^5 C_1 +^5 C_0\]

 [∵\[{}^n C_r = {}^n C_{n - r}\]]

\[\Rightarrow \sum^5_{r = 1} 5_{C_r} = 2 \times \left( \frac{5}{1} \times 4_{C_0} \right) + 2 \times \left( \frac{5}{3} \times \frac{4}{2} \times \frac{3}{1} \times 2_{C_0} \right) + 5_{C_0}\]

[∵\[{}^n C_r = \frac{n}{r} {}^{n - 1} C_{r - 1}\]]

\[\Rightarrow \sum^5_{r = 1} 5_{C_r} = 10 + 20 + 1 = 31 .\]   [∵\[{}^n C_0 = 1\]]