Question

there are 10 people standing in line for the midnight release of a new video game. if you were to mix them up, how many different lines could they make?
a 100,000 different lines
b 1024 different lines
c 3,628,800 different lines
d 100 different lines

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 10 people in a line. The formula for permutations of \( n \) distinct objects is \( n! \) (n factorial), which is \( n\times(n - 1)\times(n - 2)\times\cdots\times1 \).

Step2: Calculate \( 10! \)

We calculate \( 10! \) as follows:
\( 10! = 10\times9\times8\times7\times6\times5\times4\times3\times2\times1 \)
First, \( 10\times9 = 90 \)
Then, \( 90\times8 = 720 \)
Next, \( 720\times7 = 5040 \)
Then, \( 5040\times6 = 30240 \)
Next, \( 30240\times5 = 151200 \)
Then, \( 151200\times4 = 604800 \)
Next, \( 604800\times3 = 1814400 \)
Then, \( 1814400\times2 = 3628800 \)
Finally, \( 3628800\times1 = 3628800 \)

Wait, but looking at the options, option C is 3,628,800 different lines. So we must have made a miscalculation earlier? Wait no, 10! is indeed 3628800. Let's check the options again. Option C is 3,628,800 different lines. So that's the correct answer.

Answer:

C. 3,628,800 different lines