Question
Download Solution PDFHow many 7-letter words (with or without meaning) can be constructed using all the letters of the word CAPITAL so that all the consonants come together in each word?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
If all consonants are treated as a single unit, the problem reduces to arranging the unit (CPTL) and vowels (A, I, A).
Within the consonant unit, the consonants themselves can be permuted.
When letters repeat, the arrangements are divided by the factorial of the repeated letters.
Calculation:
⇒ Treat consonants CPTL as a single unit.
⇒ Total "super units" = {CPTL, A, I, A} = 4 units.
⇒ Arrangements of super units = 4! / 2! (since A repeats).
⇒ Arrangements of super units = 24 / 2 = 12. ...(1)
⇒ Internal arrangements of consonants CPTL = 4!.
⇒ Internal arrangements = 4 × 3 × 2 × 1 = 24. ...(2)
⇒ Total arrangements = (Arrangements of super units) × (Internal arrangements of consonants).
⇒ Total arrangements = 12 × 24.
⇒ Total arrangements = 288.
∴ The total number of 7-letter words = 288.
Hence, the correct answer is Option 3.
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