Out of 7 consonants and 4 vowels, the number of words consisting of 3 consonants and 2 vowels are

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We have a combinatorics problem. We need to find how many words can be formed using 3 consonants from 7 available consonants and 2 vowels from 4 available vowels. Our approach will be to first select the letters, then arrange them. Let's start with step 1: choosing 3 consonants from 7. We use the combination formula C of n comma r equals n factorial divided by r factorial times n minus r factorial. For C of 7 comma 3, this equals 7 factorial divided by 3 factorial times 4 factorial, which simplifies to 7 times 6 times 5 divided by 3 times 2 times 1, giving us 210 divided by 6, which equals 35 ways. Now for step 2: choosing 2 vowels from 4. Using the combination formula, C of 4 comma 2 equals 4 factorial divided by 2 factorial times 2 factorial, which simplifies to 4 times 3 divided by 2 times 1, giving us 12 divided by 2, which equals 6 ways. To find the total number of ways to select both consonants and vowels, we multiply: 35 times 6 equals 210 ways. Finally, step 3: arranging the 5 selected letters. We have 3 consonants and 2 vowels to arrange in a sequence. The number of ways to arrange 5 distinct objects is 5 factorial, which equals 5 times 4 times 3 times 2 times 1, giving us 120 arrangements. To get the final answer, we multiply the number of ways to select the letters by the number of arrangements: 210 times 120 equals 25200 words. To summarize our solution: We used combinations to select 3 consonants from 7, giving us 35 ways. We selected 2 vowels from 4, giving us 6 ways. The total letter selections equal 35 times 6, which is 210 ways. Then we arranged these 5 letters using 5 factorial, which equals 120 arrangements. Our final answer is 210 times 120, which equals 25,200 possible words.