In how many ways can 15 identical blankets be distributed among six beggars such that everyone gets at least one blanket and two particular beggars get equal blankets and another three particular beggars get equal blankets.
Text Solution
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Sol. The number of ways of distributing blankets is equal to the number of solutions of the equation 3a + 2b + c = 15
a, b, c ≥ 1, which is equal to coefficient of t 15 in
(t 3 + t 6 + t 9 . . .) (t 2 + t 4 +. . .) (t + t 2 +. .)
= coefficient of t 9 in (1+ t 3 + t 6 + t 9 ) (1+ t 2 + t 4 + t 6 + t 8 ) (1 + t + t 2 +. . . + t 9 )
(neglecting higher powers)
= coefficient of t 9 in (1 + t 2 + t 3 + t 4 + t 5 + 2t 6 + t 7 + 2t 8 + 2t 9 ) (1 + t + t 2 +….+t 9 )
= 1 + 1 + 1 + 1 +1 + 2 + 1 + 2 + 2 =12
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