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

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

hint 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⁺²C_r+1

A permutation is the non-replaceable selection of r items from a set of n items in which the order is important.
$P presuperscript n subscript r equals space fraction numerator left parenthesis n factorial right parenthesis space over denominator left parenthesis n minus r right parenthesis factorial end fraction$
A combination is created by selecting r items from a group of n items without replacing them and without regard to their order.
$C presuperscript n subscript r equals space fraction numerator left parenthesis n factorial right parenthesis space over denominator left parenthesis n minus r right parenthesis factorial cross times r factorial end fraction$
Here we have given: nCr denotes the number of combinations of n things taken r at a time, so
$C presuperscript n subscript r plus 1 end subscript plus C presuperscript n subscript r minus 1 end subscript plus 2 cross times C presuperscript n subscript r S o l v i n g space t h i s comma space w e space g e t colon C presuperscript n subscript r plus 1 end subscript plus C presuperscript n subscript r minus 1 end subscript plus C presuperscript n subscript r plus C presuperscript n subscript r N o w space u s i n g space t h e space f o r m u l a colon C presuperscript n subscript i plus C presuperscript n subscript i minus 1 end subscript equals C presuperscript n plus 1 end presuperscript subscript i A p p l y i n g space t h i s comma space w e space g e t colon C presuperscript n subscript r plus 1 end subscript plus C presuperscript n plus 1 end presuperscript subscript r A p p l y i n g space t h e space s a m e space f o r m u l a space a g a i n comma space w e space g e t colon C presuperscript n plus 2 end presuperscript subscript r plus 1 end subscript$

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 $C presuperscript n plus 2 end presuperscript subscript r plus 1 end subscript$ .

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If nCr denotes the number of combinations of n things taken r at a time, then the expression nCr+1 + nCr –1 + 2 × nCr equals-

n + 1Cr +1 n+2Cr n+2Cr+1 n+1Cr

The correct answer is: n+2Cr+1

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

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

The correct answer is: ⁿ⁺²C_r+1

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The force acting on the particle is zero at