An art gallery has 8 paintings available to display, but there are only 6 spaces (in a line) to hang them. How many different arrangements are possible? There are two correct answers; select them both.

Question

An art gallery has 8 paintings available to display, but there are only 6 spaces (in a line) to hang them.  How many different arrangements are possible?  There are two correct answers; select them both.

BlackTom AI · Accepted Answer

Permutations formula P(8,6) = 8!/(8−6)! directly applies because we are selecting and ordering 6 paintings from 8 distinct paintings.

Relevant concepts:
- Order matters (the 6 hanging spaces are in a line), so use permutations, not combinations.
- Number of ways to arrange r items from n distinct items: P(n,r) = n!/(n−r)!.
- Equivalent direct counting: choices for positions multiply: 8 choices for the first space, 7 for the second, ..., down to 3 for the sixth.

Step-by-step reasoning and calculation:
1. Use permutation formula with n = 8, r = 6:
   P(8,6) = 8! / (8−6)! = 8! / 2!.
2. Or count position-by-position:
   8 × 7 × 6 × 5 × 4 × 3.
3. Compute the product stepwise:
   8 × 7 = 56
   56 × 6 = 336
   336 × 5 = 1680
   1680 × 4 = 6720
   6720 × 3 = 20160
4. Compute factorial form to confirm:
   8! = 40320
   2! = 2
   40320 / 2 = 20160

Why the correct answers are correct:
- Both provided forms, 8!/2! and (8)(7)(6)(5)(4)(3), represent the same permutation count P(8,6) and both evaluate to 20,160, which is the number of ordered arrangements of 6 distinct paintings chosen from 8.

Why the other options are incorrect:
- 8! = 40,320 counts arrangements of all 8 paintings into 8 spots (not our case).
- 6! = 720 counts arrangements of 6 specific paintings (as if only 6 paintings exist), ignoring that there are 8 choices initially.
- (6)(5)(4)(3) = 360 is too small; it corresponds to arranging 4 of 6 items, not 6 of 8.
- Any product like 6!8!, 8!6!, or 6!2! are unrelated and do not match the correct permutation count.

Conclusion: The two correct selections are 8!/2! and (8)(7)(6)(5)(4)(3); both equal 20,160, matching the provided answers.

An art gallery has 8 paintings available to display, but there are only 6 spaces (in a line) to hang them. How many different arrangements are possible? There are two correct answers; select them both.多项选择题

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