Pascal’s triangle gives amethod for calculating the

PROBLEM 2E In designing an experiment, the researcher can often choose many different levels of the various factors in order to try to find the best combination at which to operate. As an illustration, suppose the researcher is studying a certain chemical reaction and can choose four levels of temperature, five different pressures, and two different catalysts. (a) To consider all possible combinations, how many experiments would need to be conducted? (b) Often in preliminary experimentation, each factor is restricted to two levels. With the three factors noted, how many experiments would need to be run to cover all possible combinations with each of the three factors at two levels? (Note: This is often called a 23 design.)

PROBLEM 17E A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of each of the following poker hands: (a) Four of a kind (four cards of equal face value and one card of a different value). (b) Full house (one pair and one triple of cards with equal face value). (c) Three of a kind (three equal face values plus two cards of different values). (d) Two pairs (two pairs of equal face value plus one card of a different value). (e) One pair (one pair of equal face value plus three cards of different values).

PROBLEM 1E A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, d1d2d3d4, where di, i = 1, 2, 3, 4, is selected from 1, 2, 3, 4, 5, 6, 7, and 8. How many different lock combinations are possible with such a lock?

The “eating club” is hosting a make-your-own sundae at which the following are provided: (a) How many sundaes are possible using one flavor of ice cream and three different toppings? (b) How many sundaes are possible using one flavor of ice cream and from zero to six toppings? (c) How many different combinations of flavors of three scoops of ice cream are possible if it is permissible to make all three scoops the same flavor?

PROBLEM 3E How many different license plates are possible if a state uses (a) Two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)? (b) Three letters followed by a three-digit integer? (In practice, it is possible that certain “spellings” are ruled out.)

PROBLEM 7E In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select (a) 6, 7, 8, 9. (b) 6, 7, 8, 8. (c) 7, 7, 8, 8. (d) 7, 8, 8, 8.

PROBLEM 8E How many different varieties of pizza can be made if you have the following choice: small, medium, or large size; thin ‘n’ crispy, hand-tossed, or pan crust; and 12 toppings (cheese is automatic), from which you may select from 0 to 12?

PROBLEM 9E The World Series in baseball continues until either the American League team or the National League team wins four games. How many different orders are possible (e.g., ANNAAA means the American League team wins in six games) if the series goes (a) Four games? (b) Five games? (c) Six games? (d) Seven games?

Prove \(\sum_{r=0}^{n}(-1)^{r}(n r)=0\) and \(\sum_{r=0}^{n}(n r)=2^{n}\) HINT: Consider \((1-1)^{n}\) and \((1+1)^{n}\), or use Pascal's equation and proof by induction. Equation Transcription: Text Transcription: Sum _r=0^n(-1^)r(nr)=0 Sum _r=0^n(nr)=2^n (1-1)n (1+1)n

PROBLEM 14E A bag of 36 dum-dum pops (suckers) contains up to 10 flavors. That is, there are from 0 to 36 suckers of each of 10 flavors in the bag. How many different flavor combinations are possible?

Pascal's triangle gives a method for calculating the binomial coefficients; it begins as follows: The \(n\)th row of this triangle gives the coefficients for \((a+b)^{n-1}\). To find an entry in the table other than a 1 on the boundary, add the two nearest numbers in the row directly above. The equation \(\left(\begin{array}{l}n \\r\end{array}\right)=\left(\begin{array}{c}n-1 \\ r\end{array}\right)+\left(\begin{array}{c}n-1 \\r-1\end{array}\right),\) Equation Transcription: Text Transcription: n (a+b)^n-1 (_r^n) = (_ r^n -1) + (_ r -1^n -1)

PROBLEM 11E Three students (S) and six faculty members (F) are on a panel discussing a new college policy. (a) In how many different ways can the nine participants be lined up at a table in the front of the auditorium? (b) How many lineups are possible, considering only the labels S and F? (c) For each of the nine participants, you are to decide whether the participant did a good job or a poor job stating his or her opinion of the new policy; that is, give each of the nine participants a grade of G or P. How many different “scorecards” are possible?

Prove Equation 1.2-2. Hint: First select \(n_{1}\) positions in \(\left(\begin{array}{c}n \\ n_{1}\end{array}\right)\) ways. Then select \(n_{2}\) from the remaining \(n-n_{1}\) positions in \(\left(\begin{array}{c}n-n_{1} \\ n_{2}\end{array}\right)\) ways, and so on. Finally, use the multiplication rule. Equation Transcription: Text Transcription: n_1 (_n_?^n) n_2 n-n_1 (n_2^n- n_1)

PROBLEM 16E A box of candy hearts contains 52 hearts, of which 19 are white, 10 are tan, 7 are pink, 3 are purple, 5 are yellow, 2 are orange, and 6 are green. If you select nine pieces of candy randomly from the box, without replacement, give the probability that (a) Three of the hearts are white. (b) Three are white, two are tan, one is pink, one is yellow, and two are green.

A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, d1d2d3d4, where di, i = 1, 2, 3, 4, is selected from 1, 2, 3, 4, 5, 6, 7, and 8. How many different lock combinations are possible with such a lock?

In designing an experiment, the researcher can often choose many different levels of the various factors in order to try to find the best combination at which to operate. As an illustration, suppose the researcher is studying a certain chemical reaction and can choose four levels of temperature, five different pressures, and two different catalysts. (a) To consider all possible combinations, how many experiments would need to be conducted? (b) Often in preliminary experimentation, each factor is restricted to two levels. With the three factors noted, how many experiments would need to be run to cover all possible combinations with each of the three factors at two levels? (Note: This is often called a 23 design.)

How many different license plates are possible if a state uses (a) Two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)? (b) Three letters followed by a three-digit integer? (In practice, it is possible that certain spellings are ruled out.)

The eating club is hosting a make-your-own sundae at which the following are provided: Ice Cream Flavors Toppings Chocolate Caramel Cookies n cream Hot fudge Strawberry Marshmallow Vanilla M&Ms Nuts Strawberries (a) How many sundaes are possible using one flavor of ice cream and three different toppings? (b) How many sundaes are possible using one flavor of ice cream and from zero to six toppings? (c) How many different combinations of flavors of three scoops of ice cream are possible if it is permissible to make all three scoops the same flavor?

How many four-letter code words are possible using the letters in IOWA if (a) The letters may not be repeated? (b) The letters may be repeated?

Suppose that Novak Djokovic and Roger Federer are playing a tennis match in which the first player to win three sets wins the match. Using D and F for the winning player of a set, in how many ways could this tennis match end?

In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select (a) 6, 7, 8, 9. (b) 6, 7, 8, 8. (c) 7, 7, 8, 8. (d) 7, 8, 8, 8.

How many different varieties of pizza can be made if you have the following choice: small, medium, or large size; thin n crispy, hand-tossed, or pan crust; and 12 toppings (cheese is automatic), from which you may select from 0 to 12?

The World Series in baseball continues until either the American League team or the National League team wins four games. How many different orders are possible (e.g., ANNAAA means the American League team wins in six games) if the series goes (a) Four games? (b) Five games? (c) Six games? (d) Seven games?

Pascals triangle gives a method for calculating the binomial coefficients; it begins as follows: 1 1 1 121 13 31 14 6 41 1 5 10 10 5 1 . . . . . . . . . . . . . . . The nth row of this triangle gives the coefficients for (a + b)n1. To find an entry in the table other than a 1 on the boundary, add the two nearest numbers in the row directly above. The equation  n r  =  n 1 r  +  n 1 r 1  ,called Pascals equation, explains why Pascals triangle works. Prove that this equation is correct.

Three students (S) and six faculty members (F) are on a panel discussing a new college policy. (a) In how many different ways can the nine participants be lined up at a table in the front of the auditorium? (b) How many lineups are possible, considering only the labels S and F? (c) For each of the nine participants, you are to decide whether the participant did a good job or a poor job stating his or her opinion of the new policy; that is, give each of the nine participants a grade of G or P. How many different scorecards are possible?

Prove n r=0 (1)r  n r  = 0 and n r=0  n r  = 2n. Hint: Consider (1 1)n and (1 + 1)n, or use Pascals equation and proof by induction.

A bridge hand is found by taking 13 cards at random and without replacement from a deck of 52 playing cards. Find the probability of drawing each of the following hands. (a) One in which there are 5 spades, 4 hearts, 3 diamonds, and 1 club. (b) One in which there are 5 spades, 4 hearts, 2 diamonds, and 2 clubs. (c) One in which there are 5 spades, 4 hearts, 1 diamond, and 3 clubs. (d) Suppose you are dealt 5 cards of one suit, 4 cards of another. Would the probability of having the other suits split 3 and 1 be greater than the probability of having them split 2 and 2?

A bag of 36 dum-dum pops (suckers) contains up to 10 flavors. That is, there are from 0 to 36 suckers of each of 10 flavors in the bag. How many different flavor combinations are possible?

Prove Equation 1.2-2. Hint: First select n1 positions in  n n1  ways. Then select n2 from the remaining n n1 positions in  n n1 n2  ways, and so on. Finally, use the multiplication rule.

A box of candy hearts contains 52 hearts, of which 19 are white, 10 are tan, 7 are pink, 3 are purple, 5 are yellow, 2 are orange, and 6 are green. If you select nine pieces of candy randomly from the box, without replacement, give the probability that (a) Three of the hearts are white. (b) Three are white, two are tan, one is pink, one is yellow, and two are green.

A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of each of the following poker hands: (a) Four of a kind (four cards of equal face value and one card of a different value). (b) Full house (one pair and one triple of cards with equal face value). (c) Three of a kind (three equal face values plus two cards of different values). (d) Two pairs (two pairs of equal face value plus one card of a different value). (e) One pair (one pair of equal face value plus three cards of different values).