In how many ways can 3 men and 2 women be arranged in a line if the men must be

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Question: In how many ways can 3 men and 2 women be arranged in a line if the men must be together?

Options:

Correct Answer: 120

Solution:

Treat the 3 men as one unit. So, we have 3 units (MMM, W, W). The arrangements = 3! * 3! = 36.

In how many ways can 3 men and 2 women be arranged in a line if the men must be together?

In how many ways can 3 men and 2 women be arranged in a line if the men must be together?

Correct Answer: 36

Step 1: Treat the 3 men as one single unit. This means we consider them as 'MMM'.
Step 2: Now, we have 3 units to arrange: 'MMM', 'W', and 'W'.
Step 3: Calculate the number of ways to arrange these 3 units. This is done using the factorial of the number of units, which is 3! (3 factorial).
Step 4: Calculate 3! = 3 × 2 × 1 = 6. So, there are 6 ways to arrange the units.
Step 5: Next, we need to arrange the 3 men within their unit 'MMM'. There are 3 men, so we calculate 3! for them as well.
Step 6: Calculate 3! = 3 × 2 × 1 = 6. So, there are 6 ways to arrange the men within their unit.
Step 7: Finally, multiply the number of arrangements of the units by the arrangements of the men: 6 (units) × 6 (men) = 36.

Permutations – The arrangement of objects in a specific order, considering the grouping of men as a single unit.
Factorials – The mathematical operation used to calculate the number of arrangements, represented as n! for n objects.
Grouping – The concept of treating a set of items (in this case, the 3 men) as a single unit to simplify the arrangement problem.