Too much Binomial!

Algebra Level 4

( C 0 + C 1 ) ( C 1 + C 2 ) ( C 2 + C 3 ) ⋅ ⋅ ⋅ ( C n − 1 + C n ) (C_0 + C_1)(C_1 + C_2)(C_2 + C_3)\cdot \cdot \cdot (C_{n-1} + C_n)

Let C k C_k denote the binomial coefficient ( n k ) \dbinom {n}{k} , can you simplify the product above for positive integer n > 1 n > 1 ?

C 0 C 1 C 2 C 3 ⋅ ⋅ ⋅ C n − 1 ( n + 1 ) n n ! \frac{C_0C_1C_2C_3\cdot\cdot\cdot C_{n-1}(n+1)^n}{n!} C 0 C 1 C 2 C 3 ⋅ ⋅ ⋅ C n + 1 n ! \frac{C_0C_1C_2C_3\cdot\cdot\cdot C_{n+1}}{n!} C 0 C 1 C 2 C 3 ⋅ ⋅ ⋅ C n − 1 n ! \frac{C_0C_1C_2C_3\cdot\cdot\cdot C_{n-1}}{n!} C 0 C 1 C 2 C 3 ⋅ ⋅ ⋅ C n − 2 ( n + 1 ) n n ! \frac{C_0C_1C_2C_3\cdot\cdot\cdot C_{n-2}(n+1)^n}{n!}

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Nishant Rai by Nishant Rai , 25, India

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2 solutions

Nishant Rai
Apr 20, 2015

Use the property C r n + C r − 1 n = C r n + 1 \boxed{C^n_r + C^n_{r-1} = C^{n+1}_r }

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Moderator note:

Can you elaborate on it?

On using the property C r n + C r − 1 n = C r n + 1 \boxed{C^n_r + C^n_{r-1} = C^{n+1}_r } it gets reduced to the form C 1 n + 1 C 2 n + 1 C 3 n + 1 . . . . . C n n + 1 C^{n+1}_1 C^{n+1}_2 C^{n+1}_3 ..... C^{n+1}_n

On simplification, we get

C 1 n + 1 = ( n + 1 ) ! 1 ! n ! = ( n + 1 ) C^{n+1}_1 = \frac{(n+1)!}{1! n!} = (n+1)

C 2 n + 1 = ( n + 1 ) ! 2 ! ( n − 1 ) ! = n ( n + 1 ) 2 = C 1 n ( n + 1 ) 2 C^{n+1}_2 = \frac{(n+1)!}{2!(n-1)!} = \frac{n(n+1)}{2} = \frac{C^n_1 (n+1)}{2}

C 3 n + 1 = ( n + 1 ) ! 3 ! ( n − 2 ) ! = n ( n + 1 ) ( n − 1 ) 3 ∗ 2 = C 2 n ( n + 1 ) 3 C^{n+1}_3 = \frac{(n+1)!}{3!(n-2)!} = \frac{n(n+1)(n-1)}{3*2} = \frac{C^n_2 (n+1)}{3} and so on...

Nishant Rai - 6 years, 1 month ago

Since C n = 1 C_{n} = 1 , the first and second option are technically equal.

Shak R - 6 years, 1 month ago

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Thanks, I've made an addendum.

Brilliant Mathematics Staff - 6 years, 1 month ago
Rajath Naik
Apr 21, 2015

The formula that I derived is a little bit different but serves the same purpose in the end.. nCr + nC(r+1)=(n+1)C(r+1).
But not recommended for this problem..gets too hairy in the later part.lol.but still a good result to be remembered..

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