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51.
Compute:

$space space fraction numerator left parenthesis straight n plus 1 right parenthesis space factorial over denominator left parenthesis straight n minus 1 right parenthesis space factorial end fraction$

Solution not provided.

Ans. $straight n squared plus straight n$

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52.

If $space space space fraction numerator 1 over denominator 8 factorial end fraction plus fraction numerator 1 over denominator 9 factorial end fraction space equals space fraction numerator straight x over denominator 10 factorial end fraction$ find x.

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53.

Evalaute:

$fraction numerator straight n factorial over denominator straight r factorial space left parenthesis straight n minus straight r right parenthesis factorial end fraction comma$ when

n = 5, r = 2

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54.

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55.

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56.

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57.

If $fraction numerator left parenthesis straight n minus 1 right parenthesis factorial over denominator left parenthesis straight n minus 4 right parenthesis factorial end fraction colon fraction numerator straight n factorial over denominator left parenthesis straight n minus 4 right parenthesis factorial end fraction space equals space 1 space colon space 9$

find value of n.

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58.

If $fraction numerator straight n factorial over denominator 9 factorial space left parenthesis straight n minus 9 right parenthesis factorial end fraction equals fraction numerator straight n factorial over denominator left parenthesis straight n minus 8 right parenthesis factorial space 8 factorial end fraction comma space find space value space of space fraction numerator straight n factorial over denominator 17 factorial space left parenthesis straight n minus 17 right parenthesis factorial end fraction$