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Counting non-negative integral solutions

Number of non-negative integral solutions of equation x1+x2+⋯+xn=k

= Number of ways in which k identical balls can be distributed into n distinct boxes

 (k+n-1)C(n-1)

here let x=p+2;y=q+3;z=r+4;

so we get p+q+r=17-9=8;

k=8, n=3 we get 10C2=45

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