Evaluate the following expressions:

A.) P(105,1)

B.) P(10,4)

C.) P(11,3)

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To evaluate the expressions P(105,1), P(10,4), and P(11,3), we first need to understand what the expression P(n, r) means.

P(n, r) represents the permutation of selecting r items from a total of n items without repetition and considering the order of the items. In other words, it calculates the number of ways to arrange r items from a set of n items.

Now, let's evaluate each expression step by step:

A.) P(105,1)
To calculate P(105,1), we need to find the number of ways to arrange 1 item from a set of 105 items. Since we are selecting only one item, the number of permutations will be equal to the number of items, which is 105.

So, the answer to A.) is 105.

B.) P(10,4)
For P(10,4), we have to find the number of ways to arrange 4 items from a set of 10 items.

The formula to calculate permutations is given by:
P(n, r) = n! / (n - r)!

Using this formula, we can calculate P(10,4) as follows:
P(10,4) = 10! / (10 - 4)! = 10! / 6!

Simplifying this further:
P(10,4) = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (6 * 5 * 4 * 3 * 2 * 1)

Canceling out the common terms, we are left with:
P(10,4) = 10 * 9 * 8 * 7 = 5,040

Therefore, the answer to B.) is 5,040.

C.) P(11,3)
For P(11,3), we need to find the number of ways to arrange 3 items from a set of 11 items.

Using the formula for permutations:
P(11,3) = 11! / (11 - 3)! = 11! / 8!

Simplifying this equation:
P(11,3) = (11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)

Again, canceling out the common terms, we have:
P(11,3) = 11 * 10 * 9 = 990

So, the answer to C.) is 990.