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  • For integers n and r, <img alt="\left(\begin{array}{l} n \\ r \end{array}\right)=\left\{\begin{array}{cc} { }^n C_r, & \text { if } n \geq r \geq 0 \\ 0, & \text { otherwise } \end{arr

For integers n and r, \left(\begin{array}{l} n \\ r \end{array}\right)=\left\{\begin{array}{cc} { }^n C_r, & \text { if } n \geq r \geq 0 \\ 0, & \text { otherwise } \end{array}\right.

\text { If } \sum_{i=0}^k\left(\begin{array}{c} 10 \\ i \end{array}\right)\left(\begin{array}{c} 15 \\ k-i \end{array}\right)+\sum_{i=0}^{k+1}\left(\begin{array}{c} 12 \\ i \end{array}\right)\left(\begin{array}{c} 13 \\ k+1-i \end{array}\right) exists, then k is always less than or equal to 

Option: 1

25


Option: 2

12


Option: 3

24


Option: 4

13


Answers (1)

best_answer

\sum_{i=0}^k\left(\begin{array}{c} 10 \\ i \end{array}\right)\left(\begin{array}{c} 15 \\ k-i \end{array}\right)+\sum_{i=0}^{k+1}\left(\begin{array}{c} 12 \\ i \end{array}\right)\left(\begin{array}{c} 13 \\ k+1-i \end{array}\right)\begin{aligned} =\left(\begin{array}{c} 10 \\ 0 \end{array}\right)\left(\begin{array}{c} 15 \\ k \end{array}\right) & +\left(\begin{array}{c} 10 \\ 1 \end{array}\right)\left(\begin{array}{c} 15 \\ k-1 \end{array}\right)+\ldots .+\left(\begin{array}{c} 10 \\ k \end{array}\right)\left(\begin{array}{c} 15 \\ 0 \end{array}\right) \\ & +\left(\begin{array}{c} 12 \\ 0 \end{array}\right)\left(\begin{array}{c} 13 \\ k+1 \end{array}\right)+\left(\begin{array}{c} 12 \\ 1 \end{array}\right)\left(\begin{array}{c} 13 \\ k \end{array}\right)+\ldots .+\left(\begin{array}{c} 12 \\ k+1 \end{array}\right)\left(\begin{array}{c} 13 \\ 0 \end{array}\right) \end{aligned}\begin{aligned} & =\text { Coeff. of } x^k \text { in }(1+x)^{10}(1+x)^{15} \\ & \quad+\text { Coeff. of } x^{k+1} \text { in }(1+x)^{12}(1+x)^{13} \end{aligned}

\begin{aligned} & =\text { Coeff. of } x^k \text { in }(1+x)^{25}+\text { Coeff. of } x^{k+1} \text { in }(1+x)^{25} \\ & ={ }^{25} C_k+{ }^{25} C_{k+1} \end{aligned}Now, for sum to be exists \text { s } 0 \leq k \leq 25 \text { and } 0 \leq k+1 \leq 25 \text {. }

\Rightarrow 0 \leq k \leq 24

Thus the maximum value of is 24

Posted by

Devendra Khairwa

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