Answer: 1365 ways
Explanation: this is a combination questions
n = 15
r = 4
Solution
15C⁴
= n! / r! (n-r)!
= 15! / 4! ( 15-4)!
= 15! / 4! (11)!
= 1.308 *10¹² / 24(39916800)
= 1365 ways
The number of ways four (4) members be selected for a search and rescue mission from the unit, such that at least one (1) officer is included is 650 ways.
Given the following data:
To determine the number of ways four (4) members be selected for a search and rescue mission from the unit, such that at least one (1) officer is included:
In this exercise, the number of ways four (4) members can be chosen from the fifteen (15) members without any form is restriction is given by a combination.
Mathematically, combination is given by this formula:
[tex]_nC_r = \frac{n!}{r!(n-r)!}[/tex]
Where:
Substituting the given parameters into the formula, we have;
[tex]_{15}C_4 = \frac{15!}{4!(15-4)!}\\\\_{15}C_4 = \frac{1,307,674,368,000}{24(11)!}\\\\_{15}C_4 = \frac{1,307,674,368,000}{24(39,916,800)}\\\\_{15}C_4 = \frac{1,307,674,368,000}{958,003,200}\\\\_{15}C_4 =1365 \;ways[/tex]
Next, we would determine the number of non-officers:
Non-officers = [tex]15-2=13[/tex] non-officers.
Note: The opposite of choosing team members that include at least one (1) officer is to choose a team having no officers.
Since, there are 13 non-officers, the number of ways of choosing four (4) members with no officers is given by:
[tex]_{13}C_4 = \frac{13!}{4!(13-4)!}\\\\_{13}C_4 = \frac{6,227,020,800}{24(9)!}\\\\_{13}C_4 = \frac{6,227,020,800}{24(362,880)}\\\\_{13}C_4 = \frac{6,227,020,800}{8,709,120}\\\\_{13}C_4 =715 \;ways[/tex]
Therefore, the number of ways of choosing four (4) members with at least one (1) officer is:
[tex]X = _{15}C_4 - _{13}C_4\\\\X=1365-715[/tex]
X = 650 ways.
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