Let `n and k` be positive such that `n leq (k(k+1))/2`.The number of solutions `(x_1, x_2,.....x_k), x_1 leq 1, x_2 leq 2, ........,x_k leq k`, all integers, satisfying `x_1 +x_2+.....+x_k = n`, is .......

Explanation

We have, `x_(1)+x_(2)+ . . +x_(k)=n` . . . (i)
Now, let `y_(1)=x_(1)-1,y_(2)=x_(2)-2, . . ,y_(k)=x_(k)-k`
`thereforey_(1)ge0,y_(2)ge0, . . ,y_(k)ge0`
On substituting the values `x_(1),x_(2), . . ,x_(k)` in terms of `y_(1),y_(2), . . ,y_(k)`
In Eq. (i), we get
`y_(1)+1+y_(2)+2+ . . +y_(k)+k=n`
`impliesy_(1)+y_(2)+ . . +y_(k)=n-(1+2+3+ . .+k)`
`thereforey_(1)+y_(2)+ . . +y_(k)=n-(k(k+1))/(2)=A` (say) . . . (ii)
The number of non-negative integral solutions of the Eq.
(ii) is
`=.^(k+A-1)C_(A)=((k+A-1)!)/(A!(k-1)!)`
where, `A=n-(k+1))/(2)`