Permutations And Combinations

Question
CBSEENMA11015537

Let space straight S subscript 1 space equals space sum from straight j space equals 1 to 10 of space straight j space left parenthesis negative 1 right parenthesis space to the power of 10 straight C subscript straight j space comma space straight S subscript 2 space equals space sum from straight j space equals 1 to 10 of space straight j to the power of 10 straight C subscript straight j space and space straight S subscript 3 space equals sum from straight j space equals 1 to 10 of space straight j squared space to the power of 10 straight C subscript straight j

Statement-1: S3 = 55 × 29.
Statement-2: S1 = 90 × 28 and S2 = 10 × 28.

  • Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

  • Statement-1 is true, Statement-2 is true; statement-2 is not a correct explanation for Statement-1. 

  • Statement-1 is true, Statement-2 is false.

  • Statement-1 is false, Statement-2 is true.

Solution

C.

Statement-1 is true, Statement-2 is false.

straight S subscript 1 space equals space straight capital sigma with straight i space equals space 1 below and 10 on top space straight j space left parenthesis straight j minus 1 right parenthesis fraction numerator 10 factorial over denominator straight j space left parenthesis straight j minus 1 right parenthesis left parenthesis straight j minus 2 right parenthesis factorial left parenthesis 10 minus straight j right parenthesis factorial end fraction
space equals space 90 sum from straight j space equals 2 to 10 of space fraction numerator 8 factorial over denominator left parenthesis straight j minus 2 right parenthesis factorial left parenthesis 8 minus left parenthesis straight j minus 2 right parenthesis right parenthesis factorial end fraction space equals space 90.2 to the power of 8
space and space straight S subscript 2 space equals space sum from straight i space equals 1 to 10 of fraction numerator 9 factorial over denominator straight j left parenthesis left parenthesis straight j minus 1 right parenthesis factorial space left parenthesis 9 minus left parenthesis straight j minus 1 right parenthesis right parenthesis factorial end fraction
space equals space 10 sum from straight j space equals 1 to 10 of space fraction numerator 9 factorial over denominator left parenthesis straight j minus 1 right parenthesis factorial left parenthesis 9 minus left parenthesis straight j minus 1 right parenthesis right parenthesis factorial end fraction space equals space 10.2 to the power of 9
Also comma space straight S subscript 3 space equals space 10 sum from straight j space equals 1 to 10 of space left square bracket space straight j space left parenthesis straight j minus 1 right parenthesis space plus straight j right square bracket fraction numerator 10 factorial over denominator straight j factorial left parenthesis 10 minus straight j right parenthesis factorial end fraction
space equals space sum from straight j space equals 1 to 10 of space space straight j space left parenthesis straight j minus 1 right parenthesis to the power of 10 straight C subscript straight i space equals space sum from straight j space equals 1 to 10 of space straight j to the power of 10 straight C subscript straight j
space equals space 90.2 to the power of 8 space plus space 10.2 to the power of 8
space equals space 90.2 to the power of 8 space plus space 20.2 to the power of 8
space equals space 1110.2 to the power of 8 space equals space 56.2 to the power of 8

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Some More Questions From Permutations and Combinations Chapter

Determine K, so that K + 2, 4K – 6 and 3K – 2 are three consecutive terms of an A.P.