Let n and k be positive integers such that n ≥ 2 k ( k − 1 ) . The number of solutions ( x 1 , x 2 , … x k ) , x 1 ≥ 1 , x 2 ≥ 2 , … x k ≥ k , all integers, satisfying x 1 + x 2 + ……

Question

Let n and k be positive integers such that $n \geq \frac{k ( k - 1 )}{2}$ . The number of solutions $(x_{1}, x_{2}, \dots x_{k}), x_{1} \geq 1, x_{2} \geq 2, \dots x_{k} \geq k$ , all integers, satisfying $x_{1} + x_{2} + \dots\dots + x_{k} = n$ , is $(m = \frac{2 n - k ^{2} + k - 2}{2})$ .

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Answer:

C

Solution:

Put $y_{1} = x_{1} - 1, y_{2} = x_{2} - 2 \dots, y_{k} = x_{k} - k$

On adding, $y_{1} + y_{2} - + y_{k} = n - \frac{k ( k + 1 )}{2}$ etc.

TestHub · Accepted Answer

Put y 1 ​ = x 1 ​ − 1 , y 2 ​ = x 2 ​ − 2 … , y k ​ = x k ​ − k On adding, y 1 ​ + y 2 ​ − + y k ​ = n − 2 k ( k + 1 ) ​ etc.

Mathematics - Permutation & Combination Question with Solution | TestHub

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