To determine the number of permutations of 9 things taken 3 at a time, we use the formula for permutations without repetition, which is:
P(n, k) = n! / (n - k)!
where:
n is the total number of things (9 in this case),
k is the number of things to choose at a time (3 in this case),
'!' denotes factorial, which means the product of all positive integers up to that number (for example, 5! = 5×4×3×2×1 = 120).
For our specific case:
P(9, 3) = 9! / (9 - 3)!
Now we calculate each factorial:
9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
(9 - 3)! = 6! = 6 × 5 × 4 × 3 × 2 × 1
Substitute these into the formula:
P(9, 3) = 9! / 6!
P(9, 3) = (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (6 × 5 × 4 × 3 × 2 × 1)
The 6!, which is present in both the numerator and the denominator, cancels out, so we're left with:
P(9, 3) = 9 × 8 × 7
P(9, 3) = 72 × 7
P(9, 3) = 504
Therefore, the number of permutations of 9 things taken 3 at a time is 504.
Determine the number of permutations (arrangements) possible of 9 things taken 3 at a time.
1 answer