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Home » There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.[Hint: Required number = 210 – 1].
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.[Hint: Required number = 210 – 1].
CBSE | Class 11 | Maths | NCERT Exemplar | Permutations and Combinations
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.[Hint: Required number = 210 – 1].

Solution:

It is known that,

\begin{array}{l} { }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \\ =\frac{\mathrm{n} !}{\mathrm{r} !(\mathrm{n}-\mathrm{r}) !} \end{array}

It is also known that,

\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}^{\mathrm{n}}=2^{\mathrm{n}}-1

As per the question,

10 = Number of lamps in a hall

Given,

Of the given lamps one lamp can be switched on independently

As a result, the no. of ways in which the hall can be illuminated is given by,

\begin{array}{l} \mathrm{C}_{1}^{10}+\mathrm{C}_{2}^{10}+\mathrm{C}_{3}{ }^{10}+\mathrm{C}_{4}{ }^{10}+\mathrm{C}_{5}^{10}+\mathrm{C}_{6}{ }^{10}+\mathrm{C}_{7}{ }^{10}+\mathrm{C}_{8}{ }^{10}+\mathrm{C}_{9}{ }^{10}+\mathrm{C}_{10}{ }^{10} \\ =2^{10}-1 \\ =1024-1 \\ =1023 \end{array}

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