There are 11! ways of arranging 11 unique blocks in 11 unique places, but that is not exactly what is happening here. For example, suppose they are lettered blocks. The first arrangement might be alphabetical:
A, B, C, D, E, F, G, H, I, J, K
Another arrangement might be the same thing in reverse order:
K, J, I, H, G, F, E, D, C, B, A
But that is actually a duplicate arrangement, simply viewing the same order from the opposite side. In fact, each of the 11! arrangements has been duplicated. For the correct count, divide by two.
Answers & Comments
11! = 11 x 10 x 9 x ... x 1
There are 11! ways of arranging 11 unique blocks in 11 unique places, but that is not exactly what is happening here. For example, suppose they are lettered blocks. The first arrangement might be alphabetical:
A, B, C, D, E, F, G, H, I, J, K
Another arrangement might be the same thing in reverse order:
K, J, I, H, G, F, E, D, C, B, A
But that is actually a duplicate arrangement, simply viewing the same order from the opposite side. In fact, each of the 11! arrangements has been duplicated. For the correct count, divide by two.
11! / 2 = 19,958,400
11 choices for the first block
10 choices for the second block
9 choices for the third block
...
2 choices for the tenth block
1 choice for the eleventh block.
11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
= 39,916,800 ways
11P11 = 39,916,800
11! =>
11 * 10 * 9 * ... * 3 * 2 * 1 =>
11 * 10 * 9 * 8 * 7 * 720 =>
11 * 10 * 9 * 8 * 5040 =>
11 * 10 * 9 * 40320 =>
11 * 10 * 362880 =>
3991680 * 10 =>
39,916,800