How many ways are there to choose 3 cards from a standard deck of 52 cards when order does not matter?

How many ways are there to choose 3 cards from a standard deck of 52 cards when order does not matter?

• A. 2210
• B. 22100
• C. 19600
• D. 13260

Answer: (B)

When order doesn’t matter, you count ways to choose without regard to the sequence—these are combinations. The number of ways to choose 3 cards from 52 is 52 choose 3, written as 52C3. This uses the formula nCk = n! / (k!(n−k)!). For 52 and 3, that becomes 52 × 51 × 50 divided by 3 × 2 × 1. Compute: 52 × 51 × 50 = 132,600, and 3 × 2 × 1 = 6, so 132,600 / 6 = 22,100. This is exactly the count of unordered 3-card hands. If order did matter, you’d have 52P3 = 52 × 51 × 50 = 132,600, and dividing by 3! (which is 6) gets you back to 22,100. The other numbers come from misdividing or mixing up the permutation count with the combination count. The correct result is 22,100.

When order doesn’t matter, you count ways to choose without regard to the sequence—these are combinations. The number of ways to choose 3 cards from 52 is 52 choose 3, written as 52C3. This uses the formula nCk = n! / (k!(n−k)!). For 52 and 3, that becomes 52 × 51 × 50 divided by 3 × 2 × 1. Compute: 52 × 51 × 50 = 132,600, and 3 × 2 × 1 = 6, so 132,600 / 6 = 22,100.

This is exactly the count of unordered 3-card hands. If order did matter, you’d have 52P3 = 52 × 51 × 50 = 132,600, and dividing by 3! (which is 6) gets you back to 22,100. The other numbers come from misdividing or mixing up the permutation count with the combination count. The correct result is 22,100.