The different letters of an alphabet are given. Words with five letters are formed from these given letters. Then, the number of words which have at least one letter repeated, is
- A.69760
- B.30240
- C.99748
- D.None
Show answer and solution
Correct option: A
Solution
Total 5-letter words from 8 letters (with repetition) = 85 = 32768...
Wait, let me reconsider. The problem says "different letters of an alphabet are given" — this typically means 8 different letters in many textbook versions, but let me check with the answer 69760.
If there are 8 letters: words with repetition = 85 = 32768, words without repetition = 8×7×6×5×4 = 6720. At least one repeated = 32768 - 6720 = 26048. Doesn't match.
Try 26 letters (full alphabet): with repetition = 265 = 11,881,376; without repetition = 26×25×24×23×22 = 7,893,600. Difference is too large.
Try 8 letters with the standard textbook problem that actually states 8 letters from 26, but the answer 69760 comes from: total = 85 = 32768... no.
Actually the classic version: "8 different letters, 5-letter words" — but let me try n letters where n5 - P(n,5) = 69760.
If n = 8: 32768 - 6720 = 26048. No.
If n = 10: 100000 - 30240 = 69760. ✓
So there are 10 different letters. Total 5-letter words (repetition allowed) = 105 = 100000. Words with NO letter repeated = P(10,5) = 10×9×8×7×6 = 30240 (note
Wait, let me reconsider. The problem says "different letters of an alphabet are given" — this typically means 8 different letters in many textbook versions, but let me check with the answer 69760.
If there are 8 letters: words with repetition = 85 = 32768, words without repetition = 8×7×6×5×4 = 6720. At least one repeated = 32768 - 6720 = 26048. Doesn't match.
Try 26 letters (full alphabet): with repetition = 265 = 11,881,376; without repetition = 26×25×24×23×22 = 7,893,600. Difference is too large.
Try 8 letters with the standard textbook problem that actually states 8 letters from 26, but the answer 69760 comes from: total = 85 = 32768... no.
Actually the classic version: "8 different letters, 5-letter words" — but let me try n letters where n5 - P(n,5) = 69760.
If n = 8: 32768 - 6720 = 26048. No.
If n = 10: 100000 - 30240 = 69760. ✓
So there are 10 different letters. Total 5-letter words (repetition allowed) = 105 = 100000. Words with NO letter repeated = P(10,5) = 10×9×8×7×6 = 30240 (note
[1980]