In how many ways can 7 different items be selected and arranged in a line?

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Question: In how many ways can 7 different items be selected and arranged in a line?

Options:

Correct Answer: 40320

Solution:

The number of arrangements of 7 different items is 7! = 5040.

In how many ways can 7 different items be selected and arranged in a line?

In how many ways can 7 different items be selected and arranged in a line?

Step 1: Understand that we have 7 different items to arrange.
Step 2: Realize that arranging items means we are looking for permutations.
Step 3: Know that the formula for the number of arrangements (permutations) of n different items is n! (n factorial).
Step 4: Since we have 7 items, we will use 7! (7 factorial).
Step 5: Calculate 7! by multiplying all whole numbers from 1 to 7: 7 × 6 × 5 × 4 × 3 × 2 × 1.
Step 6: Perform the multiplication: 7 × 6 = 42, then 42 × 5 = 210, then 210 × 4 = 840, then 840 × 3 = 2520, then 2520 × 2 = 5040, and finally 5040 × 1 = 5040.
Step 7: Conclude that there are 5040 different ways to arrange the 7 items.

Permutations – The question tests the understanding of permutations, specifically how to arrange a set of distinct items.