91Ó°ÊÓ

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{10} P_{4} $$

Evaluate the given binomial coefficient. $$ \left(\begin{array}{c}100 \\\2\end{array}\right) $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (3 x+y)^{3} $$

You are dealt one card from a standard 52 card deck. Find the probability of being dealt: a diamond.

Use the Fundamental Counting Principle to solve Exercises \(21-32\). In the original plan for area codes in \(1945,\) the first digit could be any number from 2 through \(9,\) the second digit was either 0 or 1, and the third digit could be any number except 0. With this plan, how many different area codes were possible?

A poker hand consists of five cards. a. Find the total number of possible five-card poker hands that can be dealt from a deck of 52 cards. b. A diamond flush consists of a five-card hand containing all diamonds. Find the number of possible five-card diamond flushes. c. Find the probability of being dealt a diamond flush.

If you are dealt 4 cards from a shuffled deck of 52 cards, find the probability that all 4 are hearts.

Use the Fundamental Counting Principle to solve Exercises \(21-32\). In the Cambridge Encyclopedia of Language (Cambridge University Press, 1987 ), author David Crystal presents five sentences that make a reasonable paragraph regardless of their order. The sentences are Mark had told him about the foxes. John looked out the window. Could it be a fox? However, nobody had seen one for months. He thought he saw a shape in the bushes. How many different five-sentence paragraphs can be formed if the paragraph begins with "He thought he saw a shape in the bushes" and ends with "John looked out of the window"?

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(x^{2}+1\right)^{16} $$

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(y^{3}-1\right)^{20} $$