Combinations If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place, be awarded If 5 sprinters compete in a race and the fastest 3 qualify for the relay team, how many different relay teams can be formed? A combination is the arrangement/selection of r objects chosen from n objects such that • the n objects are all different • no objects can be repeated • order does not make a difference (ab and ba count as one object in a combination but count as two objects in a permutation) n! • The combination formula: n Cr = r!(n − r)! • To find 8 C3 press 8 Æ MATH Æ PRB Æ nCr Æ 3 How many ways are there to play the Lotto 6/49? Lotto 6/49 requires you to choose 6 numbers from the numbers 1 through 49. Committees: How many different committees of 3 people can be formed from 7 people? How many different committees of 3 people can be formed from 7 people if the first person selected serves as the chairperson, the second as the treasurer, and the third as the secretary? If the group of 7 people consists of 3 males and 4 females, how many different committees of 3 people can be formed with 1 male and 2 females? If the group of 7 people consists of 3 males and 4 females, how many different committees of 3 people can be formed with at least one male on the committee? A box of 30 flashbulbs contains 5 defective bulbs. In how many ways can 6 bulbs be selected if: a) There is no restriction on what bulb is selected? b) Only non-defective bulbs can be selected? c) The selection must consist of 2 defective bulbs? Ten college students, including one married couple, are eligible to attend a national conference. Four students can attend, and the married couple will only go if they can both attend. How many different combinations of four students are possible? Cards: A standard deck of 52 playing cards consists of 4 suits (spaces, hearts, diamonds, and clubs) of 13 cards each. How many different 5-card hands can be formed? How many different 5-card red hands can be formed? How many different 5-card hands can be formed that consist of all hearts? How many different 5-card hands can be formed that consist of all face cards? How many different 5-card hands can be formed that consist of at least 1 heart? How many different 5-card hands can be formed that consist of exactly 3 hearts? How many different 5-card hands can be formed that consist of 3 hearts and 2 spades? How many different 5-card hands can be formed that consist of 4 of a kind? Solve: n C2 = 21 Practice 1) How many 5 card hands can be dealt from a standard deck of 52 cards if : a) All cards are red? b) 3 are red and 2 are black ? c) 4 are Aces and 1 is a King ? 2) A class has 15 boys and 12 girls. A committee of 6 must be chosen to represent the class at a meeting. In how many ways can this be done if: a) there must be 4 boys and 2 girls on the committee? b) 3 boys and 3 girls on the committee? c) the majority on the committee is girls? 3) The student council has 10 executive members, 6 of whom are girls. A committee of 5 must be chosen to plan a dance. In how many ways can the committee be chosen if: a) There must be all girls b) There must be 2 boys and 3 girls c) There must be 4 boys and 1 girl d) There must be at least 2 girls 4) From a standard deck of playing cards, how many different 5 card hands can be formed with: a) Exactly 3 black cards b) At most one black card c) At least three red cards 5) In how many ways can a committee of 4 students be chosen from a class of 7 boys and 6 girls, so that the girls will form a majority? 6) How many 3 letter combinations can be formed from the letters of the word OSAGE? 7) In five card poker, in how many ways could you be dealt: a) 4 kings b) 4 of a kind 8) A committee of 7 people is to be formed from 15 people. There are 9 males and 6 females to choose from. How many committees can be formed if: a) There are no restrictions on members b) Exactly 3 females must be on the committee c) If John (one of the boys) must be included and exactly 2 females 9) How many of the following 5 card poker hands can be created with: a) 4 black cards and 1 red card b) 3 kings and 2 queens c) 3 kings d) A flush (5 cards of the same suit) 10) A basketball coach has 5 guards and 7 forwards on his team. a) How many ways can he select a starting team of two guards and 3 forwards? b) How many different starting teams are there is the start player, who plays guard, must be included? 11) The twelve face cards are removed from a standard deck of 52 cards. From the face cards, three card hands are dealt. Determine the number of distinct 3 card hands that are possible which include: a) no restrictions b) 3 kings c) 1 queen and 2 kings d) only 1 jack 12) From a standard 52 card deck, 6 cards are dealt. How many distinct hands are possible with: a) no restrictions b) only clubs c) 2 clubs and 4 diamonds d) no sevens e) 4 tens f) only 1 jack and 4 queens 13) How many different 5 card hands can be formed with: a) at least one black card b) at least 2 kings c) at most 2 clubs d) at most 3 jacks 14) Solve a) n C2 =10 b) nn+−11C = 28 c) 222nnC − =15 HW: MC – 1,5,8,14,18,21 WR 2,3,5 Key 1a) 1b) 1c) 2a) 2b) 2c) 3) 4) (d) Since there is no way to choose an all boy committee (there are only 4 boys and 5 people are required), we can ignore that case 10CCC 5−×= 4 4 6 1 246 4 cont’) 5) 6) 53C =10 7) 8) a) 15C 7 = 6435 b) 63CC×=9 4 2520 c) ( John)××=6284 C C 1050 9) 10) 11) a) 26 CC4×= 26 1 388700 a) 5CC27× 2= 350 a) 12 C3 = 220 b) 43CC×=4 2 24 b) 11CCC×4 17×= 3 140 b) 43C = 4 c) 43482CC×=4512 c) 414CC×= 2 24 d) 41CC×=135 5148 d) 418CC×= 2 112 12a) 52C 6 = 20,358,520 12b) 13 C6 =1716 12c) 13 CC2×= 13 4 55770 12d) 48 C6 =12,271,512 12e) 44482CC× =1128 12f) 414CC×× 4441 C =176 13a) 52 CC5−= 26 5 2533180 13b) 42483CCC× +× 4 3482C+ 4 CC 4481 × =108336 13c) 39 CCCCC5+× 13 1 39 4 +× 13 2 39 3 =2357862 13d) 52CC 5− 4 4×= 48 C 2 2593320 14) n = 5 14b) n = 7 14c) n = 3 .