Question

For each of the following situations, explain why the combinations rule or the permutations rule should be used. (a) Determine the number of different groups of 5 items that can be selected from 12 distinct items. Use the combinations rule, since only the items in the group is of concern. Use the combinations rule, since the number of arrangements within each group is of interest. Use the permutations rule, since the number of arrangements within each group is of interest. Use the permutations rule, since only the items in the group is of concern. (b) Determine the number of different arrangements of 5 items that can be selected from 12 distinct items. Use the permutations rule, since the number of arrangements within each group is of interest. Use the permutations rule, since only the items in the group is of concern. Use the combinations rule, since only the items in the group is of concern. Use the combinations rule, since the number of arrangements within each group is of interest.

Answer 1

Answer:

See explanation

Step-by-step explanation:

(a) Total number of distinct objects = 12

Number of object that should be selected = 5.

Number of different groups of 5 items that can be selected from 12 distinct items

$=C^{12}_5=\dfrac{12!}{5!(12-5)!}=\dfrac{12!}{5!\cdot 7!}=\dfrac{7!\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12}{2\cdot 3\cdot 4\cdot 5\cdot 7!}=8\cdot 9\cdot 11=792$

You should use the combinations rule, since the number of arrangements within each group is of interest (the order of elements within each group is not your business).

(b) Total number of distinct objects = 12

Number of object that should be selected = 5.

Number of different arrangements of 5 items that can be selected from 12 distinct items

$=P^{12}_5=\dfrac{12!}{(12-5)!}=\dfrac{12!}{7!}=\dfrac{7!\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12}{7!}=8\cdot 9\cdot 10\cdot 11\cdot 12=95,040$

You should use the permutations rule, since the number of arrangements within each group is of interest (the order of elements within each group has value for you because arrangements 1,2,3,4,5 and 5,4,3,2,1 are different arrangements).