How many permutations can be made from 10 different things taken 4 at a time where one specific item cannot be used?
Understand the Problem
The question is asking to calculate the number of permutations that can be made from 10 different items, but one specific item cannot be included. To solve this, we first reduce the number of items to 9 (since one is excluded) and then find the number of permutations of 4 items from these 9.
Answer
$3024$
Answer for screen readers
The number of permutations of 4 items from 9 different items is $3024$.
Steps to Solve
- 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.
- 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.
- 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 $5 \times 4 \times 3 \times 2 \times 1$)
- Plug values back into the formula
Substitute the calculated factorials back into the permutation formula:
$$ P(9, 4) = \frac{9!}{5!} $$
- Simplify the expression
This can be simplified as follows (since $5!$ cancels out):
$$ P(9, 4) = 9 \times 8 \times 7 \times 6 $$
- Calculate the final result
Now calculate the product:
$$ 9 \times 8 = 72 $$
$$ 72 \times 7 = 504 $$
$$ 504 \times 6 = 3024 $$
The number of permutations of 4 items from 9 different items is $3024$.
More Information
Permutations are used in counting arrangements where the order matters. The calculation shows how many different ways we can arrange 4 items selected from 9.
Tips
- Forgetting to adjust the total number of items correctly after excluding an item.
- Mixing up permutations with combinations; remember that order matters in permutations.
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