Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels…

To count how many words can be formed by choosing letters from separate groups, first count the number of ways to SELECT the required letters from each group using the combination formula C(n, r) = n! / (r! (n − r)!). Then count the number of ways to ARRANGE all the selected letters among themselves, since a word is an ordered sequence of letters — arranging n distinct letters can be done in n! ways. By the fundamental counting principle, the total number of words = (ways to select) × (ways to arrange).

1. Select 3 consonants out of 7: C(7, 3) = (7 × 6 × 5) / (3 × 2 × 1) = 35.

2. Select 2 vowels out of 4: C(4, 2) = (4 × 3) / (2 × 1) = 6.

3. Ways to select the 5 letters (3 consonants + 2 vowels) = 35 × 6 = 210.

4. The 5 selected letters fill 5 distinct positions in the word, so they can be arranged among themselves in 5! = 5 × 4 × 3 × 2 × 1 = 120 ways.

5. By the fundamental counting principle, total number of words = 210 × 120 = 25200.

Cross-check by multiplying in a different order: 5! × C(7, 3) = 120 × 35 = 4200, then 4200 × C(4, 2) = 4200 × 6 = 25200 — the same result, confirming there is no arithmetic slip.

So, 25200 distinct words made of 3 consonants and 2 vowels can be formed.