Binomial Theorem

Question
CBSEENMA11015575

Statement space minus 1 space colon space sum from straight r space equals space 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space left parenthesis straight n plus 2 right parenthesis 2 to the power of straight n minus 1 end exponent
Statement space minus space 2 colon thin space sum from straight r equals 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r straight x to the power of straight r space equals space left parenthesis 1 plus straight x right parenthesis to the power of straight n space plus space nx space left parenthesis 1 plus straight x right parenthesis to the power of straight n minus 1 end exponent
  • Statement −1 is false, Statement −2 is true

  • 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. 

Solution

B.

Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

sum from straight r space equals 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space sum from straight r space equals space 0 to straight n of space straight r to the power of straight n straight C subscript straight r space plus space to the power of straight n straight C subscript straight r
space equals space sum from straight r space equals space 0 to straight n of space straight r space straight n over straight r space to the power of straight n minus 1 end exponent straight C subscript straight r minus 1 end subscript space plus space sum from straight r space equals space 0 to straight n of space to the power of straight n straight C subscript straight r space equals space straight n 2 to the power of straight n minus 1 end exponent space plus space 2 to the power of straight n
space equals space 2 to the power of straight n minus 1 end exponent space left parenthesis straight n plus 2 right parenthesis
Statement space minus 1 space true
sum for space of space left parenthesis straight r plus 1 right parenthesis to the power of straight n straight C subscript straight r space end subscript straight x to the power of straight r space equals space sum for space of straight r to the power of straight n space straight C subscript straight r straight x to the power of straight r space plus space sum for space of to the power of straight n straight C subscript straight r straight x to the power of straight r
space equals space straight n space sum from straight r space equals space 0 space to straight n of space to the power of straight n straight C subscript straight r minus 1 end subscript space straight x to the power of straight r space plus space sum from straight r space equals space 0 to straight n of space to the power of straight n straight C subscript straight r straight x to the power of straight r space equals space straight n space straight x space left parenthesis 1 space plus straight x right parenthesis to the power of straight n minus 1 end exponent space plus space left parenthesis 1 plus straight x right parenthesis to the power of straight n
substituting space straight x space equals space 1
sum for space of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space straight n 2 to the power of straight n minus 1 end exponent space plus space 2 to the power of straight n
Hence Statement −2 is also true and is a correct explanation of Statement −1.

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