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Prove that ((n),(r))+2((n),(r-1))+((n)...

Prove that
`((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(r))`

Text Solution

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`therefore((n),(r))=.^(n)C_(r)`
`thereforeLHS=((n),(r))+2((n),(r-1))+((n),(r-2))`
`=.^(n)C_(r)+2.^(n)C_(r-1)+.^(n)C_(r-2)`
`=(.^(n)C_(r)+.^(n)C_(r-1))+(.^(n)C_(r-1)+.^(n)C_(r-2))`
`=.^(n+1)C_(r)+.^(n+1)C_(r-1)=.^(n+2)C_(r)`
`=((n+2),(r))=RHS`
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ARIHANT MATHS-PERMUTATIONS AND COMBINATIONS -Exercise (Questions Asked In Previous 13 Years Exam)
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  8. Consider all possible permutations of the letters of the word ENDEANOE...

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  15. There are 10 points in a plane, out of these 6 are collinear. The numb...

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  17. Let n denote the number of all n-digit positive integers formed by the...

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  18. Let a(n) denote the number of all n-digit numbers formed by the digits...

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