Question

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using technology to estimate necessary permutations
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a number of children are posing for a group picture. there are five 8 - year - olds and four 9 - year - olds posing in the picture.
how many arrangements are possible where the 8 - year - olds and 9 - year - olds alternate in the lineup?
9!
5!·4!
$\frac{9!}{4!5!}$
5! + 4!

Explanation:

Step1: Determine the starting - point

Since there are five 8 - year - olds and four 9 - year - olds, the lineup must start with an 8 - year - old.

Step2: Calculate permutations of 8 - year - olds

The number of ways to arrange the five 8 - year - olds is \(5!\) (the formula for permutations of \(n\) distinct objects is \(n!\)).

Step3: Calculate permutations of 9 - year - olds

The number of ways to arrange the four 9 - year - olds is \(4!\).

Step4: Calculate total arrangements

By the multiplication principle, the total number of arrangements where they alternate is \(5!\times4!\).

Answer:

B. \(5!\cdot4!\)