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1. P(6,6) is the permutation of 6 objects taken 6 at a time. The formula for permutation is nPr = n!/(n-r)!, where n is the total number of objects and r is the number of objects taken at a time.

For P(6,6), n = 6 and r = 6, so we have:

P(6,6) = 6!/ (6-6)! = 6! / 0! = 720

Therefore, there are 720 ways to arrange 6 objects taken 6 at a time.

2. P(10,5) is the permutation of 10 objects taken 5 at a time. Using the same formula as above, we have:

P(10,5) = 10!/ (10-5)! = 10! / 5! = 30240

So, there are 30,240 ways to arrange 10 objects taken 5 at a time.

3. P(n,3) = 504. This means that there are 504 ways to arrange n objects taken 3 at a time. Using the formula for permutation, we have:

P(n,3) = n!/ (n-3)! = n x (n-1) x (n-2)

We know that P(n,3) = 504, so we can set up the equation:

n x (n-1) x (n-2) = 504

Solving for n, we get n ≈ 9.6

Therefore, there are approximately 9.6 objects to arrange taken 3 at a time to get 504 permutations.

4. P(7,r) = 840. This means that there are 840 ways to arrange 7 objects taken r at a time. Using the formula for permutation, we have:

P(7,r) = 7!/ (7-r)! = 7 x 6 x 5 x ... x (8-r)

We know that P(7,r) = 840, so we can set up the equation:

7 x 6 x 5 x ... x (8-r) = 840

Solving for r, we get r = 3

Therefore, there are 840 ways to arrange 7 objects taken 3 at a time.