MAT 112, Finite Mathematics
Worksheet #16
Counting and Probability
1. Consider a club with 12 members. Suppose 5 are boys and 7 are girls.
a). If the club elects a president and vice-president, how many different outcomes are
possible?
(12)(11) = 132 outcomes
b). If the club elects a president and vice-president, how many different outcomes are
possible if we require one of the officers to be a boy and the other a girl?
(boy president, girl VP or girl president, boy VP)
(5)(7) + (7)(5) = 70 outcomes
c). If a group of 5 members are selected for a project, how many different groups are
possible?
C 12, 5 = 792 groups
d). If a group of 5 members are selected for a project, how many different groups are
possible which include 3 boys and 2 girls?
C 5, 3 C 7, 2 = (10)(21) = 210 groups
e). If a group of 5 members are selected for a project, how many different groups are
possible which include only girls?
C 7, 5 = 21 groups
f). If a group of 5 members are selected for a project, how many different groups are
possible which include 4 or 5 girls?
C 5, 1 C 7, 4 + C 7, 5 = (5)(35) + 21 = 196 groups
g). If a group of 5 members are selected for a project, how many different groups are
possible which include exactly 2 girls or exactly 2 boys?
( That’s either 2 girls, 3 boys or 2 boys, 3 girls. )
C 5, 3 C 7, 2 + C 5, 2 C 7, 3 = (10)(21) + (10)(35) = 560 groups
2. Suppose a 5-card hand is dealt from a deck of 52 playing cards.
a). How many ways can a 5-card hand result in 5 spades?
C 13, 5 = 1287 hands
b). How many ways can a 5-card hand result in exactly 3 spades?
C 13, 3 C 39, 2 = (286)(741) = 211,926 hands
c). How many ways can a 5-card hand result in 5 non-face cards?
(There are 12 face cards and so there are 40 non-face cards. )
C 40, 5 = 658,008 hands
d). How many ways can a 5-card hand result in exactly 2 face cards?
C 12, 2 C 40, 3 = (66)(9880) = 652,080 hands
3. Suppose a 7-card hand is dealt from a deck of 52 playing cards.
a). How many 7-card hands are possible?
C 52, 7 = 133,784,560 hands
b). How many 7-card hands contain exactly 5 spades?
C 13, 5 C 39, 2 = (1287)(741) = 953,667 hands
c). How many 7-card hands include 5 or more spades?
(That’s 5 or 6 or 7 spades.)
C 13, 5 C 39, 2 + C 13, 6 C 39, 1 + C 13, 7
= (1287)(741) + (1716)(39) + 1716 =1,022,307
d). How many 7-card hands include no spades?
(Choose 7 non-spades) C 39, 7 = 15,380,937
4. Suppose your jazz collection includes 4 Yellowjackets CDs, 6 Najee CDs,
and 5 Spyro Gyra CDs.
a). If you’re asked to rank your 3 favorite CDs from this collection, how many outcomes
are possible?
P15, 3 = (15)(14)(13) = 2730 rankings
b). If you’re asked to choose your favorite CD by each of the groups above (a total of
3 CDs), how many outcomes are possible?
(4)(6)(5) = 120 outcomes
c). Suppose you take 5 of these jazz CDs along in your car. How many collections of
5 CDs are possible?
C15, 5 = 3003 groups
d). Suppose you take 5 of the CDs along in your car. How many collections of 5 CDs
are possible if they include 2 Yellowjacket CDs and 3 Najee CDs.
C4, 2 C6, 3 = (6)(20) = 120 groups
e). Suppose you take 5 of the CDs along in your car. How many collections of 5 CDs
are possible which include exactly 3 Najee CDs.
(3 Najee CDs and 2 others)
C6, 3 C9, 2 = (20)(36) = 720 groups
f). How many collections of 5 CDs are possible which include no Yellowjackets CDs?
C11, 5 = 462 groups