Set, Relations and FunctionsPYQ Dec 23Question 1907 of 217
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If B={1,2,3,4,5}\displaystyle B = \{1, 2, 3, 4, 5\}, then the number of proper subsets of B\displaystyle B is

Options

A120
B30
C31
D32
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Correct Answer

✅ Option c — 31

All Options:

  • A120
  • B30
  • C31
  • D32

Detailed Solution & Explanation

Given set B={1,2,3,4,5}\displaystyle B = \{1, 2, 3, 4, 5\}.
The number of elements in set B\displaystyle B is n=5\displaystyle n = 5.
The total number of subsets of a set with n\displaystyle n elements is given by 2n\displaystyle 2^n.
Here, the total number of subsets is:
25=322^5 = 32
A proper subset of B\displaystyle B is any subset of B\displaystyle B except B\displaystyle B itself. The number of proper subsets is given by:
Number of proper subsets=2n−1\text{Number of proper subsets} = 2^n - 1
Substituting n=5\displaystyle n = 5:
Number of proper subsets=32−1=31\text{Number of proper subsets} = 32 - 1 = 31
Hence, **Option C** is the correct answer.

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