# Evaluate the Following: 5 ∑ R = 1 5 C R - Mathematics

Evaluate the following:

$\sum^5_{r = 1} {}^5 C_r$

#### Solution

We have,

$\sum^5_{r = 1} {}^5 C_r =^5 C_1 +^5 C_2 +^5 C_3 +^5 C_4 +^5 C_5$
$\Rightarrow \sum^5_{r = 1} {}^5 C_r =^5 C_1 +^5 C_3 +^5 C_3 +^5 C_1 +^5 C_0$

[∵${}^n C_r = {}^n C_{n - r}$]

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

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

$\Rightarrow \sum^5_{r = 1} 5_{C_r} = 10 + 20 + 1 = 31 .$   [∵${}^n C_0 = 1$]
Concept: Combination
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 17 Combinations
Exercise 17.1 | Q 1.5 | Page 8