Counting and Probability - Level 1

(1) A couple plans to have three children. The birth of a boy or girl is equally

likely. What is the probability that all three children will be the same sex? Express your

answer as a common fraction.

(2) How many natural numbers between 150 and 300 are divisible by 9?

(3) Two di�erent natural numbers are selected from the set f1; 2; 3; : : : ; 6g.

What is the probability that the greatest common factor of these two numbers is one?

Express your answer as a common fraction.

(4) Simplify: 18!16!

(5) An assortment of 200 pencils is sold through a catalog for $19:90. Shipping

is an additional $6:95. Including the charges for both the pencils and the shipping, what is

the average cost, in cents, for each pencil? Express your answer to the nearest whole

number.

(6) What is the probability that if you roll two fair, standard six-sided dice, the

di�erence between the two numbers rolled will be 0? Express your answer as a common

fraction.

(7) How many integers between 200 and 300 have the sum of their digits equal

to 15?

(8) A bag contains 3 red chips, 4 blue chips and 2 white chips. If a chip is drawn

at random, what is the probability that the chip is red or blue? Express your answer as a

common fraction.

(9) These two spinners are divided into thirds and quarters, respectively. If each

of these spinners is spun once, what is the probability that the product of the results of the

two spins will be an even number? Express your answer as a common fraction.

3 4

5 5

6 7

8

(10) One hundred people were surveyed. Of these, 87 indicated they liked Mozart

and 70 indicated they liked Bach. What is the minimum number of people surveyed who

could have said they liked both Mozart and Bach?

(11) A survey was conducted regarding the number of automotive vehicles owned

per household. The results of the survey are as follows:Vehicles Households

0 3

1 9

2 10

3 2

4 1

Express the average number of vehicles per household as a decimal rounded to the nearest

tenth.

(12) The arithmetic mean of four di�erent two-digit numbers is 60. What is the

greatest possible value of any of these four numbers?

(13) Florida has license plates with an alpha-numeric format like the one pictured.

North Dakota, on the other hand, has a di�erent format, also pictured. Assuming all 10

digits are equally likely to appear in the numeric positions, and all 26 letters are equally

likely to appear in the alpha positions, how many more license plates can Florida issue than

North Dakota?

HJF 94KFlorida

DGT 317North Dakota

(14) A �ve-digit number is called a mountain number if the �rst three digits

increase and the last three digits decrease. For example, 35; 763 is a mountain number but

35; 663 is not. How many �ve-digit numbers greater than 70; 000 are mountain numbers?

(15) What is the sum of all the positive two-digit integers where one of the digits

is four times the other?

(16) How many integers satisfy the inequality -39 < x < 39?

(17) Jamie rolled a standard six-sided die that has each face painted either red,

green or blue. In 100 rolls, the die landed with a red face up 50 times, with a blue face up

17 times, and with a green face up 33 times. Based on this data, how many faces of the

die would you expect to be green?

(18) Thomas took three 100-point quizzes, and the mean score for the three

quizzes was 87 points. If the highest score is taken away, the mean of the remaining two

scores would be 83:5 points. If the lowest score is taken away, the mean of the remaining

two scores would be 89 points. What is the highest score?

(19) What is the probability of guessing the right answer to a multiple choice

question if the choices are a, b, c, d, and e?

(20) A suitcase lock has 3 dials with the digits 0; 1; 2; :::; 9 on each. How many

di�erent settings are possible if all three digits have to be di�erent?

(21) On a survey, students must give exactly one of the answers provided to each

of these three questions:

a) Were you born before 1990? (Yes / No)

b) What is your favorite color? (Red / Green / Blue / Other)

c) Do you play a musical instrument? (Yes / No)

How many di�erent answer combinations are possible?

(22) Of the numbers 710; 45and 3

4which number is the arithmetic mean of the

other two?

(23) Interior numbers begin in the third row of Pascal's Triangle. The sum of the

interior numbers in the fourth row is 6. The sum of the interior numbers of the �fth row is

14. What is the sum of the interior numbers of the seventh row?

(24) The digits 1, 3 and 5 are each used once to form each of the possible

three-digit positive integers. The three-digit integers are listed from greatest to least.

Which integer is listed �fth?

(25) The positive three-digit integer N has a ones digit of 0. What is the

probability that N is divisible by 4? Express your answer as a common fraction.

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Answer Sheet

Number Answer Problem ID

1 1/4 B5D2

2 17 numbers 25122

3 11/15 5113

4 306 1A1A

5 13 cents 5431

6 1/6 CA32

7 6 integers 0AC1

8 79

B2151

9 2/3 4CD

10 57 people 34C01

11 1.6 30512

12 99 B552

13 28,121,600 DB4B

14 36 B213

15 165 D0C

16 77 integers 02C

17 2 faces D0012

18 94 points 33CB

19 1/5 01C12

20 720 settings BCB2

21 16 combinations 4DD2

22 34

0DC41

23 62 3BA51

24 153 1ABB

25 1/2 A4C01

Copyright MATHCOUNTS Inc. All rights reserved

Solutions

(1) 1/4 ID: [B5D2]

The �rst child can be a boy or a girl. In either case, the second child has a 12probability of

being the same gender as the �rst child. Same for the third child, whose has a 12probability

of being the same gender as the �rst child. So, the probability of all three children being

the same sex is 12� 12=

1

4.

(2) 17 numbers ID: [25122]

For a number to be divisible by 9, the sum of its digits must also be divisible by 9. Thus,

the least and greatest numbers in the rang from 150 to 300 that are divisible by nine are

153 and 297. Since starting from 153, every ninth consecutive integer is also divisible by

nine, we have that there are 297�1539

= 1449

= 16 integers greater than 153 but in the

designated range that are divisible by nine. Counting 153 as well, there are 17 numbers

that meet the requirements.

(3) 11/15 ID: [5113]

We consider all the two-element subsets of the six-element set f1; 2; 3; 4; 5; 6g. There are(6

2

)= 15 such subsets. And of these, only the subsets f2; 4g; f2; 6g; f3; 6g; f4; 6g are not

relatively prime. So the probability of the two-element subset's elements having greatest

common factor one is 1� 415

=11

15.

(4) 306 ID: [1A1A]

We can express 18!16!

as 18�17�16!16!

. To simplify this expression, we can cancel 16! from the

numerator and denominator, leaving us with 18 � 17, or 306 .

(5) 13 cents ID: [5431]

The average cost per pencil is equal to the total cost divided by the number of pencils. The

total cost is 19:90 + 6:95 = 26:85 dollars, or 2685 cents, and there are 200 pencils. Thus,

the average cost is 2685200

� 13 cents.

(6) 1/6 ID: [CA32]

If the di�erence between the two rolls is 0, then the rolls must be the same. Regardless of

what the �rst roll is, the second roll is the same with1

6probability.

(7) 6 integers ID: [0AC1]

Because the integers are between 200 and 300, we know that the hundreds digit is a 2.

Thus, we are looking for two digits that sum to 15� 2 = 13. There are three such pairs of

digits: 6 and 7, 5 and 8, and 4 and 9. Each pair yields two possibilities for creating a

three-digit integer, since the order matters. Thus, there are 3 � 2 = 6 such integers.

(8) 79

ID: [B2151]

There are 3 + 4 + 2 = 9 chips in all. There are 3 + 4 = 7 chips that are red or blue out of

the 9 chips, so the probability of drawing a red or blue chip is7

9.

(9) 2/3 ID: [4CD]

We will subtract the probability that the product is odd from 1 to get the probability that

the product is even. In order for the product to be odd, we must have both numbers be

odd. There are 2 � 2 = 4 possibilities for this (a 3 or 5 is spun on the left spinner and a 5 or

7 on the right) out of a total of 3 � 4 = 12 possibilities, so the probability that the product

is odd is 4=12 = 1=3. The probability that the product is even is 1� 1=3 = 2=3 .

(10) 57 people ID: [34C01]

The minimum number of people who like both Mozart and Bach is achieved when the

number of people who like Mozart but not Bach is maximized. There are 100� 70 = 30

people who do not like Bach. If all these people like Mozart, then the number of people

who like Bach and Mozart is 87� 30 = 57 .

(11) 1.6 ID: [30512]

We divide the total number of vehicles by the total number of households:

0(3) + 1(9) + 2(10) + 3(2) + 4(1)

3 + 9 + 10 + 2 + 1=

39

25= 1:56 � 1:6

(12) 99 ID: [B552]

One number will be as large as possible when the other three are as small as possible. Since

the integers are all di�erent, the three smallest integers are 10, 11, and 12. The remaining

number is (60 � 4)� 10� 11� 12 = 207. However, this is not a two-digit number; it is

clear, though, that by making the three other integers larger, we can get the fourth down

to any desired two-digit number. The largest possible two-digit number is 99 , which can

be achieved, for example with the other three numbers 30, 31, and 80.

(13) 28,121,600 ID: [DB4B]

Florida issues license plates in which the �rst three and last slots are �lled with letters, and

the fourth and �fth are �lled with digits. Thus, there are 264 � 102 Florida license plates

possible. North Dakota, however, issues license plates in which the �rst three slots are �lled

with letters and the last three slots are �lled with digits. There are thus 263 � 103 possible

North Dakota license plates. Multiplying these out and taking the di�erence yields an

answer of 28121600 .

(14) 36 ID: [B213]

The �rst three digits must be 7, 8, 9 for them to be increasing. For the last two digits, we

can choose any two distinct digits less than 9, place the larger one in the tens place and

the smaller one in the units place, and obtain a mountain number. There are 92= 36 ways

to choose two digits less than 9, so the answer is 36 .

(15) 165 ID: [D0C]

The only such possible two-digit integers will be composed with either a 1 and a 4 or a 2

and an 8. So, we simply want to compute 14 + 41 + 28 + 82. Adding the �rst two and the

last two in pairs gives 55 + 110, which is easily computed to be 165 .

(16) 77 integers ID: [02C]

These may be broken into three sets: the positive numbers 1 to 38, of which there are 38;

the negative numbers �1 to �38, of which there are 38; and 0. So the answer is

38 + 38 + 1 = 77

(17) 2 faces ID: [D0012]

If n faces are painted green, we would expect approximately n=6 of the rolls to result in

green. Since approximately 1=3 of the 100 rolls resulted in green, we would expect n = 2 .

(18) 94 points ID: [33CB]

Let the three scores be a, b, and c , and assume without loss of generality that a � b � c .

Since the mean score is 87, we have the equation a+b+c

3= 87, so a + b + c = 261. When

the highest score c is taken away, the mean becomes 83.5: a+b

2= 83:5) a + b = 167.

Subtracting the second equation from the �rst, we have

(a + b + c)� (a + b) = 261� 167) c = 94. Thus, the highest score is 94 points. (We

did not have to use the additional information about the mean when the lowest score is

taken away.)

(19) 1/5 ID: [01C12]

There are �ve choices total and only one of them is correct, so the probability of guessing

the correct answer is1

5.

(20) 720 settings ID: [BCB2]

There are 10 possibilities for the �rst digit. After the �rst digit has been chosen, there are

9 possibilities for the second digit, and after the �rst two digits have been chosen there are

8 possibilities for the last digit. The total number of possible settings is 10 � 9 � 8 = 720 .

(21) 16 combinations ID: [4DD2]

The �rst question has 2 answer choices, the second 4 answer choices, and the third 2. So

there are 2� 4� 2 = 16 di�erent answer sets.

(22) 34

ID: [0DC41]

The arithmetic mean must be between the other two numbers, so we �nd the middle

quantity by putting the fractions in a comparable form. We have 710; 45= 8

10; 34= 7:5

10. The

middle quantity is 7:510, so the arithmetic mean is

3

4. Our answer makes sense since 7:5 is

the arithmetic mean of 7 and 8.

(23) 62 ID: [3BA51]

If Pascal's Triangle begins with row zero, then the sum of the elements in row n is 2n. The

interior numbers refer to all numbers in the row except the 1 at each end, so the sum of

the interior elements in row n is 2n � 1� 1 = 2n � 2. For the fourth row, n = 3 and the

sum is 23 � 2 = 6. For the �fth row, n = 4 and the sum is 24 � 2 = 14. So for the seventh

row, n = 6 and the sum is 26 � 2 = 64� 2 = 62 .

(24) 153 ID: [1ABB]

There are 3! = 6 possible three-digit integers. So the �fth number on the list will be the

second smallest. The two smallest integers have 1 as the hundreds digit. The smallest is

135; the second-smallest is 153 .

(25) 1/2 ID: [A4C01]

An integer is divisible by 4 if and only if a number formed from the last two digits is

divisible by 4. If the units digit is 0, all the numbers with even tens digits are divisible by 4

(00, 20, 40, 60, 80), and all the numbers with odd tens digits are not (10, 30, 50, 70, 90).

Since there are the same number of even digits as odd digits, there is a 1=2 probability

that N is divisible by 4.

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