5.4.52 A grocery store makes up fruit baskets using as many as 4 apples, 3
pears, and four oranges. A basket must contain at least one piece of fruit.
How many different fruit selections are possible?


Solution:
(#apples)x(#pears)x(#oranges) - 1 = 5x4x5 - 1 = 99. (Note that there are
5 choices for the number of apples, etc, since 0 is a possibility. The
subtraction of 1 excludes the case of an empty "fruit basket")
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5.6.8 In the world series the American league team ("A") and the National
league team ("N") play until one team wins 4 games. If the sequence of
winners is designated by letters (NAAAA means the National league won the
first game and lost the next 4), how many different sequences are possible?

Solution: First count number of series won by A:

i) 4 game series (AAAA): 1 possibility only.

ii) 5 game series (NAAAA, ANAAA, AANAA, AAANA): 4 possibilities. (Note that
AAAAN is not included since that series ended in 4 games.)

iii)6 game series:

	a) N wins 2 in first 4: C(4,2) = 6 (A must win games 5 and 6.)
	b) N wins 1 in first 4 and game 5: C(4,1) = 4.

A total of 6+4 = 10 6 game series.

iv)7 game series:

	a)N wins 3 in first 5: C(5,3) = 10 (A must win games 6 and 7.)
	b)N wins 2 in first 5 and game 6: C(5,2) = 10

A total of 10+10 = 20 7 game series.

Total number of series won by A: 1+4+10+20=35

Total number of series = 2x35 = 70.

Have all possibilities actually occured? No. One student points out that
no team has ever come back from 3 games down.

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Supplementary Exercises

11: Out of a group of 115 applicants for jobs at the World Bank, 70 speak
French, 65 speak Spanish, 65 speak German, 43 speak French and Spanish,
35 speak Spanish and German, and 35 speak all 3 languages. How many of the 
people speak none of the languages?


Using a 3 set (G,F, and S) Venn diagram shows that the answer is 0. 

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23: A survey at a small New England college showed that 400 students skied,
300 played ice hockey, and 150 did both. How many students participated in
at least one of these sports?

N(H union S) = n(H) + n(S) - n(H intersect S) = 400+300-150 = 550.

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26: How many 7 digit numbers are even and have a 3 in the hundreds place?


Solution: The 7th digit of 7 digit even number must be 0,2,4,6,or 8. There is
only 1 choice for the 5th digit and 10 choices for each of the others
except for the first digit which cannot be 0. Thus there are

9x10x10x10x1x10x5 = 450,000

possible such numbers.

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41: In how many ways can 5 people be assigned to seats in a 12 seat room?

Solution: P(12,5) = 95,040. (Pick a seat for person 1, a different seat
for person 2, ...)