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Question: The sum \(\sum_{i = 0}^{m}{\begin{pmatrix} 10 \\ i \end{pmatrix}\begin{pmatrix} 20 \\ m - i \end{pma...

The sum ∑i=0m(10i)(20m−i),(Where(pq)=0ifp<q)\sum_{i = 0}^{m}{\begin{pmatrix} 10 \\ i \end{pmatrix}\begin{pmatrix} 20 \\ m - i \end{pmatrix},\left( Where\begin{pmatrix} p \\ q \end{pmatrix} = 0ifp < q \right)}, is

maximum when m is

A

5

B

15

C

10

D

20

Answer

20

Explanation

Solution

For m = 5, ∑i=05(10i)(205−i)\sum_{i = 0}^{5}{\begin{pmatrix} 10 \\ i \end{pmatrix}\begin{pmatrix} 20 \\ 5 - i \end{pmatrix}}

= (100)(205)+(101)(204)+......+(105)(200)\begin{pmatrix} 10 \\ 0 \end{pmatrix}\begin{pmatrix} 20 \\ 5 \end{pmatrix} + \begin{pmatrix} 10 \\ 1 \end{pmatrix}\begin{pmatrix} 20 \\ 4 \end{pmatrix} + ...... + \begin{pmatrix} 10 \\ 5 \end{pmatrix}\begin{pmatrix} 20 \\ 0 \end{pmatrix},

For m = 10, ∑i=010(10i)(2010−i)\sum_{i = 0}^{10}{\begin{pmatrix} 10 \\ i \end{pmatrix}\begin{pmatrix} 20 \\ 10 - i \end{pmatrix}}

= (100)(2010)+(101)(209)+(102)(208)......+(1010)(200)\begin{pmatrix} 10 \\ 0 \end{pmatrix}\begin{pmatrix} 20 \\ 10 \end{pmatrix} + \begin{pmatrix} 10 \\ 1 \end{pmatrix}\begin{pmatrix} 20 \\ 9 \end{pmatrix} + \begin{pmatrix} 10 \\ 2 \end{pmatrix}\begin{pmatrix} 20 \\ 8 \end{pmatrix}...... + \begin{pmatrix} 10 \\ 10 \end{pmatrix}\begin{pmatrix} 20 \\ 0 \end{pmatrix},

For m = 15, ∑i=015(10i)(2015−i)\sum_{i = 0}^{15}{\begin{pmatrix} 10 \\ i \end{pmatrix}\begin{pmatrix} 20 \\ 15 - i \end{pmatrix}}

= (100)(2015)+(101)(2014)+(102)(2013)......+(1010)(205)\begin{pmatrix} 10 \\ 0 \end{pmatrix}\begin{pmatrix} 20 \\ 15 \end{pmatrix} + \begin{pmatrix} 10 \\ 1 \end{pmatrix}\begin{pmatrix} 20 \\ 14 \end{pmatrix} + \begin{pmatrix} 10 \\ 2 \end{pmatrix}\begin{pmatrix} 20 \\ 13 \end{pmatrix}...... + \begin{pmatrix} 10 \\ 10 \end{pmatrix}\begin{pmatrix} 20 \\ 5 \end{pmatrix}

and for m = 20, ∑i=020(10i)(2020−i)\sum_{i = 0}^{20}{\begin{pmatrix} 10 \\ i \end{pmatrix}\begin{pmatrix} 20 \\ 20 - i \end{pmatrix}}

= (100)(2020)+(101)(2019)+.....+(1010)(2010)\begin{pmatrix} 10 \\ 0 \end{pmatrix}\begin{pmatrix} 20 \\ 20 \end{pmatrix} + \begin{pmatrix} 10 \\ 1 \end{pmatrix}\begin{pmatrix} 20 \\ 19 \end{pmatrix} + ..... + \begin{pmatrix} 10 \\ 10 \end{pmatrix}\begin{pmatrix} 20 \\ 10 \end{pmatrix}

Clearly, the sum is maximum for m = 15.

Note that 10Cr is maximum for r = 5 and 20Cr is maximum for r = 10. Note that the single term 10C5 x 20C10 (in case m = 15) is greater than the sum 10C0 20C2 + 10C9 20C1 + 10C10 20C0 (in case m = 10).

Also the sum incase m = 10 is same as that in case m = 20.