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Probability Level pending

If n C r + 3 n C r − 1 + 3 n C r − 2 + n C r − 3 = n + k C r ^{ n }{ C }_{ r }+3^{ n }{ C }_{ r-1 }+3^{ n }{ C }_{ r-2 }+^{ n }{ C }_{ r-3 }=^{ n+k }{ C }_{ r } for every n , r ϵ N n,r\epsilon N r ϵ [ 3 , n + 3 ] r\epsilon [3,n+3] . Find k 4 − 1 { k }^{ 4 }-1 .

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The answer is 80.

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Abhishek Sharma by Abhishek Sharma , 22, India

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1 solution

U Z
Dec 21, 2014

n C r + n C r + 1 = n + 1 C r + 1 \boxed{^{ n }{ C }_{ r } + ^{ n }{ C }_{ r + 1 } = ~ ^{ n + 1}{ C }_{ r + 1}}

= n C r + n C r − 1 + 2 ( n C r − 1 + n C r − 2 ) + n C r − 2 + n C r − 3 = ~ ^{ n }{ C }_{ r } + ^{ n }{ C }_{ r - 1 } + 2( ^{ n }{ C }_{ r - 1} + ^{ n }{ C }_{ r - 2} ) + ^{ n }{ C }_{ r - 2} + ^{ n }{ C }_{ r - 3}

= n + 1 C r + 2 ( n + 1 C r − 1 ) + n + 1 C r − 2 = ~^{ n + 1 }{ C }_{ r} + 2( ^{ n + 1 }{ C }_{ r - 1}) + ^{ n + 1}{ C }_{ r - 2}

= n + 1 C r + n + 1 C r − 1 + n + 1 C r − 1 + n + 1 C r − 2 = ~ ^{ n + 1 }{ C }_{ r} + ^{ n + 1 }{ C }_{ r - 1} + ^{ n + 1 }{ C }_{ r - 1} + ^{ n + 1 }{ C }_{ r - 2}

= n + 2 C r + n + 2 C r − 1 = ~ ^{ n + 2 }{ C }_{ r} + ^{ n + 2 }{ C }_{ r - 1}

= n + 3 C r = ~ ^{ n + 3 }{ C }_{ r}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly.! Elegant and Compact.

Abhishek Sharma - 6 years, 5 months ago

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