In how many ways can 5 children be arranged in 4 chairs?
Understand the Problem
The question is asking us to calculate the number of ways 5 children can be arranged in 4 chairs. This is a permutation problem because the order of the children matters. We need to find the number of permutations of 5 children taken 4 at a time.
Answer
$120$
Answer for screen readers
$120$
Steps to Solve
- Identify the problem as a permutation
Since the order in which the children are seated matters, this is a permutation problem. We need to find the number of ways to arrange 5 children in 4 chairs.
- Apply the permutation formula
The number of permutations of $n$ items taken $r$ at a time is given by the formula: $$P(n, r) = \frac{n!}{(n-r)!}$$ where $n!$ (n factorial) is the product of all positive integers up to $n$.
- Plug in the values
In our case, $n = 5$ (the number of children) and $r = 4$ (the number of chairs). So we have: $$P(5, 4) = \frac{5!}{(5-4)!}$$
- Simplify the expression
$$P(5, 4) = \frac{5!}{1!}$$
- Calculate the factorial
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ $1! = 1$
- Divide to get the final answer
$$P(5, 4) = \frac{120}{1} = 120$$
$120$
More Information
There are 120 different ways to arrange 5 children in 4 chairs.
Tips
A common mistake is to confuse permutations and combinations. Permutations are used when the order matters, while combinations are used when the order does not matter. In this problem, the order in which the children are seated is important, so we must use permutations. Another common mistake is incorrectly calculating the factorial.
AI-generated content may contain errors. Please verify critical information
