The given expression is $P(n, 3) = 90$, which represents the number of permutations of $n$ objects taken 3 at a time. The formula for permutations is:

$P(n, r) = \frac{n!}{(n - r)!}$

In your case, $r = 3$, so the formula becomes:

$P(n, 3) = \frac{n!}{(n - 3)!}$

You're given that $P(n, 3) = 90$, so we can set up the equation:

$\frac{n!}{(n - 3)!} = 90$

Now, expand the factorial terms:

$\frac{n(n-1)(n-2)(n-3)!}{(n-3)!} = 90$

The $(n-3)!$ terms cancel out, leaving:

$n(n - 1)(n - 2) = 90$

Now, solve for $n$:

1. Multiply the terms:

$n(n - 1)(n - 2) = 90$

2. Try values for $n$ to find the solution:

• For $n = 6$: $6 \times 5 \times 4 = 120 \quad (\text{too high})$

• For $n = 5$: $5 \times 4 \times 3 = 60 \quad (\text{too low})$

• For $n = 7$: $7 \times 6 \times 5 = 210 \quad (\text{too high})$

By trial and error or simplifying, you’ll find that $n = 10$. This gives:

[ 10 \times 9 \times 8 = 720 \ (wrong step check)} ]