Sigma with combination

Probability Level 3

∑ r = 0 10 r ( 10 r ) 3 r ( − 2 ) 10 − r = ? \large \sum_{r=0}^{10} r {10 \choose r} 3^r(-2)^{10-r} = \, ?

Notation: ( n k ) = n ! k ! ( n − k ) ! \dbinom nk = \dfrac{n!}{k!(n-k)!} denotes the binomial coefficient .

10 30 20 300

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Atul Kumar Ashish by Atul Kumar Ashish , 21, India

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1 solution

Chew-Seong Cheong
Oct 20, 2016

S = ∑ r = 0 10 r ( 10 r ) 3 r ( − 2 ) 10 − r = ∑ r = 1 10 r ( 10 r ) 3 r ( − 2 ) 10 − r = ∑ r = 1 10 r ⋅ 10 r ⋅ ( 9 r − 1 ) 3 r ( − 2 ) 10 − r = 10 ∑ r = 0 9 ( 9 r ) 3 r + 1 ( − 2 ) 9 − r = 30 ( 3 − 2 ) 9 = 30 \begin{aligned} S & = \sum_{r=0}^{10} r {10 \choose r} 3^r (-2)^{10-r} \\ & = \sum_{r=\color{#D61F06}{1}}^{10} r {\color{#3D99F6}{{10 \choose r}}} 3^r (-2)^{10-r} \\ & = \sum_{r=1}^{10} r \cdot {\color{#3D99F6}{\frac {10}r \cdot {9 \choose r-1}}} 3^r (-2)^{10-r} \\ & = 10 \sum_{r=\color{#D61F06}0}^{\color{#D61F06}9} {9 \choose \color{#D61F06}r} 3^{\color{#D61F06}{r+1}} (-2)^{\color{#D61F06}{9-r}} \\ & = 30 (3-2)^9 \\ & = \boxed{30} \end{aligned}

10 Helpful 1 Interesting 3 Brilliant 0 Confused

Nice use of color to explain the changes you made in each step!

Eli Ross Staff - 4 years, 7 months ago

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Saving explanation in words.

Chew-Seong Cheong - 4 years, 7 months ago

Wonderfully explained!

Michael Ng - 4 years, 7 months ago

Great Explanation.

Tauhid Khan Tamim - 3 months, 3 weeks ago

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