Permutation and Combination

Find the number of ways of permuting the numbers 1,2,3,......upto n . So that they are first increasing and then decreasing ....

#MathProblem

Note by Arijit Banerjee
7 years, 7 months ago

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Comments

Say, the position of nn is kthk^{th} from beginning.

k−1k-1 numbers would be in left ofnn, and n−kn - k numbers would be to its right, Select k−1k-1 numbers in the left in(n−1k−1){{n-1} \choose {k-1}} ways and arrange them in one order(increasing). Now the others would automatically be arranged in right of nn in descending order in 11 way.

Hence , we conclude that :

No. of ways =∑k=2n−1(n−1k−1)=2n−1−2\displaystyle \sum_{k = 2}^{n - 1} {{n-1} \choose {k-1}} = \boxed{2^{n-1} - 2}

jatin yadav - 7 years, 7 months ago
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