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Question
Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
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Solution
Total number of things = n
3 things must be together
∴ The number of remaining things = n – 3
Number of things to be selected = r
Out of r, 3 are always together
∴ Number of ways of selection = `""^(n - 3)"C"_(r - 2)`
Now permutation of 3 things which are always together = 3!
Number of permutations of (r – 2) things = (r – 2)!
∴ Total number of arrangements = `""^(n - 3)"C"_(r - 2) xx (r - 2)! xx 3!`
Hence the required arrangements = `""^(n - 3)"C"_(r - 2) xx (r - 2)! xx 3!`
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