How many words can be formed by using 3 letters from the word “DELHI” ?

Concept: When forming an ordered arrangement (a “word”) using r items chosen from n distinct items, and a different order counts as a different outcome, the number of ways is the permutation nPr = n! / (n - r)!.

Application: The word “DELHI” has 5 distinct letters -- D, E, L, H, I -- and a different letter order gives a different 3-letter word, so this is a permutation problem with n = 5 and r = 3.

1. Write the permutation formula for n = 5 and r = 3: nPr = 5! / (5 - 3)!.

2. Simplify the denominator: 5! / (5 - 3)! = 5! / 2!.

3. Expand the factorials: 5! = 5 x 4 x 3 x 2 x 1 = 120 and 2! = 2 x 1 = 2.

4. Divide to get the count: 120 / 2 = 60.

Cross-check: Counting directly confirms this -- there are 5 choices for the first letter of the word, 4 remaining choices for the second letter, and 3 remaining choices for the third letter, so 5 x 4 x 3 = 60, matching the result from the formula.