Set, Relations and FunctionsMTP June 24 Series IIIQuestion 1940 of 217
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The number of proper subsets of the set {3,4,5,6,7}\displaystyle \{3, 4, 5, 6, 7\} is

Options

A32
B31
C30
D25
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Correct Answer

✅ Option b — 31

All Options:

  • A32
  • B31
  • C30
  • D25

Detailed Solution & Explanation

Let the given set be S={3,4,5,6,7}\displaystyle S = \{3, 4, 5, 6, 7\}.
The number of elements in set S\displaystyle S is n(S)=5\displaystyle n(S) = 5.
The total number of subsets of a set with n\displaystyle n elements is 2n\displaystyle 2^n. For n=5\displaystyle n = 5, the number of subsets is 25=32\displaystyle 2^5 = 32.
A proper subset of a set S\displaystyle S is any subset of S\displaystyle S other than the set S\displaystyle S itself.
Therefore, the number of proper subsets of S\displaystyle S is given by:
Number of proper subsets=2n−1\text{Number of proper subsets} = 2^n - 1
Number of proper subsets=25−1=32−1=31\text{Number of proper subsets} = 2^5 - 1 = 32 - 1 = 31
Hence, **Option B** is the correct answer.

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