How many permutations can be made from 10 different things taken 4 at a time where one specific item cannot be used?

Steps to Solve

1. Identify the total items available after exclusion

Since one item is excluded from the initial 10 items, we now have 9 items available for selection.

2. Use the permutation formula

The formula for permutations of $r$ items from $n$ items is given by:

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

Here, $n = 9$ and $r = 4$. We can substitute these values into the formula.

3. Calculate the factorial

Calculate the factorials involved in the permutation formula:

• Calculate $9!$ (which is $9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$)
• Calculate $(9-4)! = 5!$ (which is $5 \times 4 \times 3 \times 2 \times 1$)

4. Plug values back into the formula

Substitute the calculated factorials back into the permutation formula:

$$ P(9, 4) = \frac{9!}{5!} $$

5. Simplify the expression

This can be simplified as follows (since $5!$ cancels out):

$$ P(9, 4) = 9 \times 8 \times 7 \times 6 $$

6. Calculate the final result

Now calculate the product:

$$ 9 \times 8 = 72 $$

$$ 72 \times 7 = 504 $$

$$ 504 \times 6 = 3024 $$