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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

Verified by Experts

`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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