Question: The sum $ \sum\limits _{i=0}^{m}{( \begin{matrix} 10 \\ i \\ \end{matrix} )}( \begin{matrix} 20 \\ m-i \\ \end{matrix} ), $ $ ( where( \begin{matrix} p \\ q \\ \end{matrix} )=0ifp<q ) $ , is maximum when m is [IIT Screening 2002]
Options:
A) 5
B) 15
C) 10
D) 20
Show Answer
Answer:
Correct Answer: B
Solution:
- For m $ =5,\sum\limits _{i=0}^{5}{( \begin{aligned} & 10 \\ & i \\ \end{aligned} )( \begin{aligned} & 20 \\ & 5-i \\ \end{aligned} )} $ $ =( \begin{aligned} & 10 \\ & 0 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 5 \\ \end{aligned} )+( \begin{aligned} & 10 \\ & 1 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 4 \\ \end{aligned} )+…+( \begin{aligned} & 10 \\ & 5 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 0 \\ \end{aligned} ), $ for m = 10, $ \sum\limits _{i=0}^{10}{( \begin{aligned} & 10 \\ & i \\ \end{aligned} )( \begin{aligned} & 20 \\ & 10-i \\ \end{aligned} )} $ $ =( \begin{aligned} & 10 \\ & 0 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 10 \\ \end{aligned} )+( \begin{aligned} & 10 \\ & 1 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 9 \\ \end{aligned} )+( \begin{aligned} & 10 \\ & 2 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 8 \\ \end{aligned} ) $ $ +…+( \begin{aligned} & 10 \\ & 10 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 0 \\ \end{aligned} ) $ , for m = 15, $ \sum\limits _{i=0}^{15}{( \begin{aligned} & 10 \\ & i \\ \end{aligned} )( \begin{aligned} & 20 \\ & 15-i \\ \end{aligned} )} $ $ =( \begin{aligned} & 10 \\ & 0 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 15 \\ \end{aligned} )+( \begin{aligned} & 10 \\ & 1 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 14 \\ \end{aligned} )+( \begin{aligned} & 10 \\ & 2 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 13 \\ \end{aligned} )+..+( \begin{matrix} 10 \\ 10 \\ \end{matrix} )( \begin{matrix} 20 \\ 5 \\ \end{matrix} ) $ and for m = 20, $ \sum\limits _{i=0}^{20}{( \begin{aligned} & 10 \\ & i \\ \end{aligned} )( \begin{aligned} & 20 \\ & 20-i \\ \end{aligned} )} $ $ =( \begin{aligned} & 10 \\ & 0 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 20 \\ \end{aligned} )+( \begin{aligned} & 10 \\ & 1 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 19 \\ \end{aligned} )+…+( \begin{aligned} & 10 \\ & 10 \\ \end{aligned} )( \begin{aligned} & 20 \\ & 10 \\ \end{aligned} ) $ Clearly, the sum is maximum for m = 15. Note that $ ^{10}C _{r} $ is maximum for r = 5 and $ ^{20}C _{r} $ is maximum for r = 10. Note that the single term $ ^{10}C_5\times {{}^{20}}C _{10} $ (in case m = 15) is greater than the sum $ ^{10}C_0{{}^{20}}C _{10}+{{}^{10}}C_1{{}^{20}}C_9+{{}^{10}}C_2{{}^{20}}C_8+….. $ $ ^{10}C_8{{}^{20}}C_2+{{}^{10}}C_9{{}^{20}}C_1+{{}^{10}}C _{10}{{}^{20}}C_0 $ (in case m = 10). Also the sum in case m = 10 is same as that in case m = 20.
BETA
