In how many ways can 15 different books be di | vedclass

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(B) To divide $15$ different books into $5$ equal groups,each group will contain $15 / 5 = 3$ books.First,we calculate the number of ways to distribut

Vedclass · Accepted Answer

$1401400$

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(B) To divide $15$ different books into $5$ equal groups,each group will contain $15 / 5 = 3$ books.First,we calculate the number of ways to distribute the books into $5$ distinct groups of $3$ each:Number of ways $= {}^{15}C_{3} 	imes {}^{12}C_{3} 	imes {}^{9}C_{3} 	imes {}^{6}C_{3} 	imes {}^{3}C_{3}$.Since the order of the $5$ groups does not matter (they are identical sets),we must divide by $5!$ to remove the permutations of the groups.Required number of ways $= \frac{{}^{15}C_{3} 	imes {}^{12}C_{3} 	imes {}^{9}C_{3} 	imes {}^{6}C_{3} 	imes {}^{3}C_{3}}{5!}$.Calculating the combinations:${}^{15}C_{3} = \frac{15 	imes 14 	imes 13}{3 	imes 2 	imes 1} = 455$.${}^{12}C_{3} = \frac{12 	imes 11 	imes 10}{3 	imes 2 	imes 1} = 220$.${}^{9}C_{3} = \frac{9 	imes 8 	imes 7}{3 	imes 2 	imes 1} = 84$.${}^{6}C_{3} = \frac{6 	imes 5 	imes 4}{3 	imes 2 	imes 1} = 20$.${}^{3}C_{3} = 1$.Total $= \frac{455 	imes 220 	imes 84 	imes 20 	imes 1}{120} = \frac{168168000}{120} = 1401400$.

In how many ways can $15$ different books be divided equally among $5$ sets or groups?

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