# 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 ....

Note by Arijit Banerjee
4 years, 8 months ago

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Say, the position of $$n$$ is $$k^{th}$$ from beginning.

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

Hence , we conclude that :

No. of ways =$$\displaystyle \sum_{k = 2}^{n - 1} {{n-1} \choose {k-1}} = \boxed{2^{n-1} - 2}$$

- 4 years, 8 months ago