How many ways are there to choose 2 items from a set of 5 without regard to order?

• A. 5
• B. 15
• C. 10
• D. 20

Answer: (C)

Choosing 2 items from 5 without regard to order uses combinations. The number of ways to choose k items from n is given by n choose k, computed as n!/(k!(n−k)!). For n = 5 and k = 2, that’s 5!/(2!3!) = 120/(2×6) = 10. You can also list all unordered pairs: there are 10 distinct pairs like (item1, item2), (item1, item3), (item1, item4), (item1, item5), (item2, item3), (item2, item4), (item2, item5), (item3, item4), (item3, item5), and (item4, item5). Since order doesn’t matter, each pair is counted once (for example, (A,B) and (B,A) are the same). So the correct count is 10.