This exercise concerns 5-card poker hands from a standard 52-card deck. How many hands contain a royal straight flush (that is, the 10, jack, queen, king, ace of one suit)?

Prove or disprove the given statement. 297 is a composite number.

Prove the given statement. The sum of two odd integers is even.

Use mathematical induction to prove that the statements are true for every positive integer n. $1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n+1} n^{2}=\frac{(-1)^{n+1}(n)(n+1)}{2}$

A royal flush consists of an ace, king, queen, jack, and 10 all in the same suit. If 7 cards are dealt at random from a standard deck of 52 cards, determine the probability of getting a (a) Royal flush in spades. (b) Royal flush in any suit.

Add, subtract, and multiply in binary: (a) 1111 and 1010 (b) 110110 and 11101 (c) 100100 and 10110

Prove the given statement. Prove that $\sqrt{3}$ is not a rational number.

Using predicate logic, prove that each argument is valid. Use the predicate symbols shown. Every farmer owns a cow. No dentist owns a cow. Therefore no dentist is a farmer. F(x), C(x), O(x, y), D(x)

Use the Ratio Test to determine whether each series converges or diverges or state it provides no information. $\sum_{k=1}^{\infty}(-1)^{k} \frac{k}{e^{k}}$

14 copies of a code module are to be executed in parallel on identical processors organized into two communicating clusters, A and B. Cluster A contains 16 processors and cluster B contains 32 processors. How many different sets of processors can be used if all modules are to be executed either on cluster A or on cluster B?

14 copies of a code module are to be executed in parallel on identical processors organized into two communicating clusters, A and B. Cluster A contains 16 processors and cluster B contains 32 processors. How many different sets of processors can be used?

14 copies of a code module are to be executed in parallel on identical processors organized into two communicating clusters, A and B. Cluster A contains 16 processors and cluster B contains 32 processors. How many different sets of processors can be used if cluster A has 3 failed processors and cluster B has 2 failed processors?

Deal with the following situation: Of a company’s personnel, 7 people work in design, 14 in manufacturing, 4 in testing, 5 in sales, 2 in accounting, and 3 in marketing. A committee of 6 people is to be formed to meet with upper management. How many committees with 1 member from each department are possible?

A palindrome is a string of characters that reads the same forward and backward. How many five-letter English language palindromes are possible?