Cool Problem In The Hot Summer
The letters of the word COCHIN are permutated and all permutations are arranged in an alphabetical order as in an English dictionary.
What is the number of words that appear before the word COCHIN?
Clarification: The words formed don't actually have to make sense, CCOHNI is an example.
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1 solution
In alphabetical order, the letters are CCHINO.
COCHIN is the first possible permutation beginning CO. As such, the permutations that come before COCHIN are exactly those matching these patterns C C ( H I N O ) C H ( C I N O ) C I ( C H N O ) C N ( C H I O ) where unbracketed portions are constant, and bracketed portions denote arbitrary permutations of the letters contained.
Each bracketed portion contains four letters, all distinct, the number of permutations of which is 4 ! = 2 4 . Since there are four disjoint patterns to match, each of this form, the total number of matching permutations is 4 × 4 ! = 9 6 .