8.2
Decide whether the exercise involves permutations or combinations, and then solve the problem. A bag contains 12 black, 1 red, and 3 yellow jelly beans; you take 3 at random. How many samples are possible in which the jelly beans are the following. (i) Does the problem involve permutations or combinations? (a) In how many ways can 3 jelly beans be chosen so all are black? (b) In how many ways can 3 jelly beans be chosen so all are red? (c) In how many ways can 3 jelly beans be chosen so all are yellow? (d) In how many ways can 3 jelly beans be chosen so that 2 are black and 1 is red? (e) In how many ways can 3 jelly beans be chosen so that 2 are black and 1 is yellow? (f) In how many ways can 3 jelly beans be chosen so that 2 are yellow and 1 is black? (g) In how many ways can 3 jelly beans be chosen so that 2 are red and 1 is yellow
(i) Combinations a) C(12,30) = 220 b) none, only 1 bean is red c) C(3,3) = 1 d) C(12,2)*C(1,1) = 66 e) C(12,5)*C(3,1) = 792 f) C(3,2)*C(12,1) = 36 g) none, there is only 1 red
Decide whether the exercise involves permutations or combinations, and then solve the problem. There are 9 rotten apples in a crate of 29 apples. (i) Does the problem involve permutations or combinations? (a) How many samples of 3 apples can be drawn from the crate? (b) How many samples of 3 could be drawn in which all 3 are rotten? (c) How many samples of 3 could be drawn in which there are two good apples and one rotten one?
(i) combinations a) C(29,3) = 3654 b) C(9,3) = 84 n in this case is the # of rotten apples c) C(10,1)*C(20,2) = 1710 First, find how many apples are good. It is known that there are 29 apples total and 9 are rotten. Subtract to find how many are good. 29−9=20 Identify n and r for the two separate combinations. The number of samples of 4 good apples can be found using C(20,2) = 190 and the number of samples of 1 rotten apple can be found using C(9,1) = 9 9*190 = 1710
In how many ways can a hand of 10 clubs be chosen from an ordinary deck?
C(13,10) 13=#number of clubs in deck
From a group of 17 smokers and 21 nonsmokers, a researcher wants to randomly select 7 smokers and 7 nonsmokers for a study. In how many ways can the study group be selected?
C(17,7) * C(21,7) = 2,261,413,440 ways multiply combinations of smokers by combinations of non smokers
How many different 11-card hands can be selected from an ordinary deck?
C(52,11)
From a group of 14 newly hired office assistants, 5 are selected. Each of these 5 assistants will be assigned to a different manager. In how many ways can they be selected and assigned? Since the order does matter, use permutations.
P(14,5) = 240240 ways
A certain website requires users to log on using a security password. (a) If passwords must consist of six letters followed by a single digit, determine the total number of possible distinct passwords. (b) If passwords must consist of six non-repetitive letters followed by a single digit, determine the total number of possible distinct passwords.
a) (26^6)(10) = 3,089,157,760 b) 26*25*24*23*22*21*20*10 = 1,657,656,000
A company sells regular hamburgers as well as a larger burger. Either type can include cheese, relish, lettuce, tomato, mustard, or ketchup. a. How many different hamburgers can be ordered with exactly 4 extras? b. How many different regular hamburgers can be made that use any 4 of the extras? c. How many different regular hamburgers can be ordered with at least 3 extras?
a) 2* C(6,6) = 2 - multiply by 2 since there are 2 different sizes b) 1*C(6,6) = 1 c) 1*( C(6,6) + C(6,5) ) = 7
Six orchids from a collection of 23 are to be selected for a flower show. Complete parts (a) and (b) below. (a) In how many ways can this be done? (b) How many ways can the 6 be selected if 3 special plants from the 23 must be included?
a) C(23,6) = 100947 b) C(20,3) = 1140 - subtract 3 from n and r
A lottery game requires that you pick 7 different numbers from 1 to 65. If you pick all 7 winning numbers, you win the jackpot. Complete parts (a) and (b) below. (a) How many ways are there to choose 7 numbers if order is not important? Choose the correct answer below. (b) How many ways are there to choose 7 numbers if order matters?
a) C(65,7) = 696,190,560 b) P(65,7) = 3,508,800,422,400
A group of 8 workers decides to send a delegation of 4 to their supervisor to discuss their grievances. Complete parts (a) through (c) below. (a) How many delegations are possible? (b) If it is decided that a particular worker must be in the delegation, how many different delegations are possible? (c) If there are 3 women and 5 men in the group, how many delegations would include at least 1 woman?
a) C(8,4) = 70 b) C(7,3) = 35 Since one person must be in thedelegation, there will be one less person to choose from and one less spot to fill. To find this, subtract 1 from n and r and then find the combination. c) a+b+c+d = answer C(4,4)*C(3,0) = a C(4,3)*C(3,1) = b C(4,2)*C(3,2) = c C(4,1)*C(3,3) = d
