How many ways can 4 men and 3 women be arranged in a line if the men must be tog

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Question: How many ways can 4 men and 3 women be arranged in a line if the men must be together? (2019)

Options:

Correct Answer: 5040

Exam Year: 2019

Solution:

Treat the 4 men as one unit. Thus, we have 4 units (1 man unit + 3 women) to arrange: 4! * 4! = 24 * 24 = 576.

How many ways can 4 men and 3 women be arranged in a line if the men must be together? (2019)

How many ways can 4 men and 3 women be arranged in a line if the men must be together? (2019)

Step 1: Treat the 4 men as one single unit or block. This means instead of 4 individual men, we now have 1 'man block'.
Step 2: Now, we have to arrange this 'man block' along with the 3 women. So, we have a total of 4 units to arrange: 1 man block + 3 women.
Step 3: Calculate the number of ways to arrange these 4 units. The formula for arranging n units is n!. Here, n = 4, so we calculate 4! (which is 4 factorial).
Step 4: Calculate 4! = 4 × 3 × 2 × 1 = 24. This is the number of ways to arrange the 4 units.
Step 5: Now, we need to arrange the 4 men within their block. The number of ways to arrange 4 men is also 4! = 24.
Step 6: To find the total arrangements, multiply the number of ways to arrange the 4 units by the number of ways to arrange the 4 men: 24 (arrangements of units) × 24 (arrangements of men) = 576.
Step 7: Therefore, the total number of ways to arrange 4 men and 3 women in a line, with the men together, is 576.

Permutations – The arrangement of objects in a specific order, considering the grouping of men as a single unit.
Grouping – Treating multiple items as a single unit to simplify the arrangement process.