First slide

Combinations

Question

There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which hail can be illuminated.

Easy

A

$2^{10} - 2$
B

$2^{10} - 1$
C

$2^{10} + 1$
D

None of these

Solution

Total number of ways
$\begin{array}{l} =^{10} C_{1} +^{10} C_{2} +^{10} C_{3} +^{10} C_{4} +^{10} C_{5} +^{10} C_{6} + \dots +^{10} C_{10} \\ = 2^{10} - 1 [∵^{n} C_{0} +^{n} C_{1} +^{n} c_{2} + \dots = 2^{n}] \end{array}$

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