An instructor gives an exam with fifteen questions. Students are allowed to choose any eleven to answer. (a) How many different choices of eleven questions are there?   (b) Suppose six questions require proof and nine do not. (i) How many groups of eleven questions contain four that require proof and seven that do not?   (ii) How many groups of eleven questions contain at least one that requires proof?   (iii) How many groups of eleven questions contain at most three that require proof?   (c) Suppose the exam instructions specify that at most one of the questions 1 and 2 may be included among the eleven. How many different choices of eleven questions are there?   (d) Suppose the exam instructions specify that either both questions 1 and 2 are to be included among the eleven or neither is to be included. How many different choices of eleven questions are there?

Q: Ten pirates find a sack of gold coins. When they try to divide up the gold (with equal shares for…

A:

Q: Suppose we want to choose 5 letters, without replacement, from 15 distinct letters. (a) How many…

A: Solution-: Here, n=15,r=5 We want to choose 5 letter, without replacement from 15 distinct letter.…

Q: In how many different ways could the medals be awarded such that Arthur does not receive a medal?

A: Solution: Total number of athletes participated to marathon = 100 Arthur is one of the participated…

Q: How many seven-digit numbers can be formed with the first two digits odd and the last five digits…

A: The total number of digits from 0 to 9 is 10.

Q: n how many ways can 5 men, 4 women, and 3 children be arranged in a row of 12 seats if a. the men…

A: Given that  Number of men = 5 Number of women = 4 Number of children = 3 Number of seats in the row…

Q: (4) What formula did you use for question three, and why? (5) You know that Person A, Person B,…

A: (4) To state and explain: The formula used for question three. (5) It is known that Person A, Person…

Q: (a) How many ways can a student choose eight out of ten questions to answer on an exam? (b) How many…

A: (a) total number of question=10 to find number of ways can a student choose eight out of ten  we…

Q: How many different ways can 5 out of 8 different shirts, 4 out of 10 different pairs of pants, and 3…

A: We have given that,  Shirts = 8 , pairs of pants  = 10 and pairs of socks = 6

Q: Suppose we want to choose 2 objects, without replacement, from the 5 objects pencil, eraser, desk,…

A: a)There is total 5 objects.

Q: A president and vice president for a club must be chosen from four people. How many different…

A: The permutation is defined as the arrangement of things or objects into a definite order. In…

Q: dont understand the question.

A:

Q: How many three-letter "words" (some may be nonsense words) can be made from 4 letters "FGHI" if…

A: Permutation and combination used to calculate the number of possible ways to arrange or select a…

Q: Suppose we want to choose 2 letters, without replacement, from the 5 letters A, B, C, D, and E. (If…

A: given data,  we want  to choose 2 letters, ,without replacement  letters A,B,C,D  and Ewe have to…

Q: For how many n = {1, 2, ..., 695} is n a multiple of one a multiple of one or more of 8, 3, or 7?

A: In the range of [1, 695] the number of numbers divisible by an integer k =  represents the floor…

Q: For how many n = {1, 2,…, 645} is n a multiple of one or more of 8, 7, or 3?

A: n belongs to (1,2,....645) is n a multiple of one or more of 8,7 or 3.

Q: In a survey of 400 members of a local sports club, 200 members indicated that they plan to attend…

A: As per bartleby guidelines only first three subparts are to be solved. Please upload other parts…

Q: Use the formula for „P, to solve the following question. A club with nine members is to choose three…

A: Introduction: In mathematics, a permutation of a set is a loose organisation of its members into a…

Q: Jack packs 3 pairs of shoes, 4 pairs of boots, 8 pairs of jeans, 9 pairs of dress pants, and 2 dress…

A: He can pick a shoe in 3 ways, a boot in 4 ways, a jeans in 8 ways, a dress pant in 9 ways, a dress…

Q: A math instructor has 35 calculators in her office. Of them, 17 have dead batteries, 13 have cracked…

A:

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.