In most state and local courts, 12 jurors and 2 alternates are chosen from a pool of 30 prospective jurors.The order of the alternates is specified. If a juror is unable to serve, then the first alternate will replace the juror.The second alternate will be called on if another juror is dismissed.

a. In how many ways can 12 jurors and first and second alternates be chosen from 30 people.
b. In federal court cases, 12 jurors and 4 alternates are usually selected form a pool of 64 prospective jurors. In how many ways can this be done?

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a. To find the number of ways to choose 12 jurors and first and second alternates from a pool of 30 people, we need to use the concept of permutations.

The number of ways to choose the first juror is 30 because any of the 30 people can be the first juror.
After the first juror is chosen, there are 29 people left from which to choose the second juror.
Similarly, after the second juror is chosen, there are 28 people left from which to choose the third juror, and so on, until we have chosen 12 jurors.

Therefore, the number of ways to choose 12 jurors from 30 people is calculated using the concept of permutations:

30P12 = 30! / (30 - 12)!
= 30! / 18!
= 30 * 29 * 28 * ... * 19 * 18!

Next, we need to consider the alternates. Since they have a specific order as specified, we multiply the number of ways to choose the jurors by the number of ways to choose the alternates.

For the first alternate, there are 18 people left from which to choose (since 12 jurors have been chosen).
After the first alternate is chosen, there are 17 people left from which to choose the second alternate.

Therefore, the number of ways to choose the alternates is:

18P1 * 17P1 = 18 * 17 = 306

Finally, we multiply the number of ways to choose the jurors by the number of ways to choose the alternates:

30P12 * 18P1 * 17P1 = (30 * 29 * 28 * ... * 19 * 18!) * (18 * 17) = 306 * (30 * 29 * 28 * ... * 19)
= 306 * 30!

So, there are 306 * 30! ways to choose 12 jurors and first and second alternates from a pool of 30 people.

b. Similarly, for federal court cases, we have a pool of 64 prospective jurors from which to choose 12 jurors and 4 alternates.

Using the same approach as above, the number of ways to choose 12 jurors from 64 people is:

64P12 = 64! / (64 - 12)!
= 64! / 52!
= 64 * 63 * 62 * ... * 53 * 52!

The number of ways to choose the alternates is:

52P4 = 52! / (52 - 4)!
= 52! / 48!
= 52 * 51 * 50 * 49 * 48!

Therefore, the total number of ways to choose 12 jurors and 4 alternates is:

64P12 * 52P4 = (64 * 63 * 62 * ... * 53 * 52!) * (52 * 51 * 50 * 49 * 48!)

This calculation will give you the answer for part b.

Note: The calculation involves large numbers, so using a calculator or computer program to perform the actual calculations is recommended.

a. 145422675

b. 4.885*10^14