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# If ^{n}C_{r} denotes the number of combinations of n things taken r at a time, then the expression ^{n}C_{r+1} + ^{n}C_{r –1} + 2 × ^{n}C_{r} equals-

^{n + 1}C_{r +1}
^{n+2}C_{r}
^{n+2}C_{r+1}
^{n+1}C_{r}

^{n + 1}C_{r +1}^{n+2}C_{r}^{n+2}C_{r+1}^{n+1}C_{r}Hint:

### By choosing some items from a set and creating subsets, permutation and combination are two approaches to represent a group of objects. It outlines the numerous configurations for a particular set of data. Permutations are the selection of data or objects from a set, whereas combinations are the order in which they are represented. Here we have given nCr denotes the number of combinations of n things taken r at a time, we have to find the expression nCr+1 + nCr –1 + 2 × nCr.

## The correct answer is: ^{n+2}C_{r+1}

### A permutation is the non-replaceable selection of r items from a set of n items in which the order is important.

A combination is created by selecting r items from a group of n items without replacing them and without regard to their order.

Here we have given: nCr denotes the number of combinations of n things taken r at a time, so

The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .

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