Let $A = \{a, b, c, d, e\}$ then the number of proper subsets is
(A) 31
(B) 32
(C) 30
(D) 29

Question

Accepted Answer

> [!TLDR]
> Number of proper subsets of an $n$-element set $= 2^n - 1$; for $n=5$ that is $31$.

### Step 1 — Count the elements
$A = \{a, b, c, d, e\}$ has $n = 5$ elements.

### Step 2 — Total number of subsets
> [!FORMULA]
> $$	ext{Total subsets} = 2^n = 2^5 = 32$$

### Step 3 — Remove the set itself for PROPER subsets
A proper subset is any subset that is not equal to the set itself.
$$	ext{Proper subsets} = 2^n - 1 = 32 - 1 = 31$$

**Why the other options are wrong:** (B) 32 counts all subsets (includes $A$ itself); (C) 30 wrongly removes two subsets; (D) 29 removes three. …

Step 1 — Count the elements

Step 2 — Total number of subsets

Step 3 — Remove the set itself for PROPER subsets

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