Explanation Given: m jobs to be assigned to n employees. Calculation: The number of ways to assign m jobs to n employees so that each employee gets at least one job is same as the number of onto functions from a set of m elements to a set of n elements, for a onto function, If m < n then there will be no function working from A onto B , so assume m ≥ n . From the principle of inclusion-exclusion, There are n m functions of all kinds from A to B . If b ∈ B , there are ( n − 1 ) m functions from A to B . Functions whose ranges do not include b are needed to subtracted from n m so C ( n , 1 ) ( n − 1 ) m will be subtracted from n m Hence n m − C ( n , 1 ) ( n − 1 ) m . When a function whose range misses two members of B gets subtracted twice in that computation and it should be subtracted only once so functions whose ranges miss at least two points should be added back in B . If b 0 , b 1 ∈ B there are ( n − 2 ) m functions from A to B \ { b 0 , b 1 } and there are C ( n , 2 ) pairs of points of B . So, this should be added back in C ( n , 2 ) ( n − 2 ) m to get n m − C ( n , 1 ) ( n − 1 ) m + C ( n , 2 ) ( n − 2 ) m Similarly If b 0 , b 1 , ... , b n ∈ B there are ( − 1 ) n − 1 C ( n , n − 1 ) ⋅ 1 m functions from A to B \ { b 0 , b 1 , To determine (b) The number of ways to assign 7 jobs to 3 employees so that each employee is assigned at least one job.

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

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Chapter 8, Problem 14RQ

To determine

(a)

To determine the number of ways to assign m jobs to n employees so that each employee is assigned at least one job.

To determine

(b)

The number of ways to assign 7 jobs to 3 employees so that each employee is assigned at least one job.

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Chapter 8, Problem 13RQ

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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

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(a) To determine the number of ways to assign m jobs to n employees so that each employee is assigned at least one job.

Solution Summary: The author calculates the number of ways to assign m jobs to n employees so that each employee gets at least one job.