Permutations and Combinations 3 Question 12

12. Let n and k be positive integers such that nk(k+1)2. The number of solutions (x1,x2,,xk), x11,x22,,xkk for all integers satisfying x1+x2++xk=n is …

(1996, 2M)

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Answer:

Correct Answer: 12. 12(2nk2+k2)

Solution:

  1. The number of solutions of x1+x2++xk=n

= Coefficient of tn in (t+t2+t3+)(t2+t3+)

= Coefficient of tn in t1+2++k(1+t+t2+)k

(tk+tk+1+)

Now, 1+2++k=k(k+1)2=p

[say]

and 1+t+t2+=11t

Thus, the number of required solutions

= Coefficient of tnp in (1t)k

= Coefficient of tnp in [1+kC1t+k+1C2t2+k+2C3t3+]

=k+np1Cnp=rCnp

where, r=k+np1=k+n112k(k+1)

=12(2k+2n2+k2k)=12(2nk2+k2)



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