Exercise 1

How many different combinations of management can there be to fill the positions of president, vice-president and treasurer of a football club knowing that there are 12 eligible candidates?

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Exercise 2

How many different ways can the letters in the word "micro" be arranged if it always has to start with a vowel?

Exercise 3

How many combinations can the seven colors of the rainbow be arranged into groups of three colors each?

Exercise 4

How many different five-digit numbers can be formed with only odd numbered digits? How many of these numbers are greater than 70,000?

Exercise 5

How many games will take place in a league consisting of four teams? (Each team plays each other twice, once at each teams respective "home" location)

Exercise 6

10 people exchange greetings at a business meeting. How many greetings are exchanged if everyone greets each other once?

Exercise 7

How many five-digit numbers can be formed with the digits 1, 2 and 3? How many of those numbers are even?

Exercise 8

How many lottery tickets must be purchased to complete all possible combinations of six numbers, each with a possibility of being from 1 to 49?

Exercise 9

How many ways can 11 players be positioned on a soccer team considering that the goalie cannot hold another position other than in goal?

Exercise 10

How many groups can be made from the word "house" if each group consists of 3 alphabets?

Exercise 11

Sarah has 8 colored pencils that are all unique. She wants to pick three colored pencils from her collection and give them to her younger sister. How many different combinations of colored pencils can Sarah make from 8 pencils?

Exercise 12

Alice has 6 chocolates. All of the chocolates are of different flavors. She wants to give two of her chocolates to her friend. How many different combinations of chocolates can Alice make from six chocolates?

 

Solution of exercise 1

How many different combinations of management can there be to fill the positions of president, vice-president and treasurer of a football club knowing that there are 12 eligible candidates?

The order of the elements does matter.

The elements cannot be repeated.

P(12, 3) = 12 \cdot 11\cdot 10 = 1320

 

Solution of exercise 2

How many different ways can the letters in the word "micro" be arranged if it always has to start with a vowel?

The words will begin with i or o followed by the remaining 4 letters taken from 4 by 4.

The order of the elements does matter.

The elements cannot be repeated.

2 P_4 = 2 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 48

 

Solution of exercise 3

How many combinations can the seven colors of the rainbow be arranged into groups of three colors each?

The order of the elements does not matter.

The elements cannot be repeated.

\binom{7}{3} = \frac {7!} {3! (7-3!)}

\binom{7}{3} = \frac {7!} {3! \cdot 4!)} = 35

 

Solution of exercise 4

How many different five-digit numbers can be formed with only odd numbered digits? How many of these numbers are greater than 70,000?

The order of the elements does matter.

The elements cannot be repeated.

n = 5      k = 5

P_5 = 5! = 120

The odd numbers greater than 70,000 have to begin with 7 or 9. Therefore:

2 P_4 = 2 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 48

 

Solution of exercise 5

How many games will take place in a league consisting of four teams? (Each team plays each other twice, once at each teams respective "home" location)

The order of the elements does matter.

The elements cannot be repeated.

P(4, 3) = 4 \cdot 3 = 12

 

Solution of exercise 6

10 people exchange greetings at a business meeting. How many greetings are exchanged if everyone greets each other once?

The order of the elements does not matter.

The elements cannot be repeated.

C_{10} ^ 2 = \frac {10 \cdot 9 } {2} = 45

 

Solution of exercise 7

How many five-digit numbers can be formed with the digits 1, 2 and 3? How many of those numbers are even?

The order of the elements does matter.

The elements are repeated.

P(3, 5) = 3^5 = 243

If the number is even it can only end in 2.

P(3, 4) = 3^4 = 81

 

Solution of exercise 8

How many lottery tickets must be purchased to complete all possible combinations of six numbers, each with a possibility of being from 1 to 49?

The order of the elements does not matter.

The elements cannot be repeated.

\binom{49}{6} = \frac {49!} {6! (49-6!)}

\binom{49}{6} = 13983816

 

Solution of exercise 9

How many ways can 11 players be positioned on a soccer team considering that the goalie cannot hold another position other than in goal?

Therefore, there are 10 players who can occupy 10 different positions.

The order of the elements does matter.

The elements cannot be repeated.

10! = 3628800

Solution of exercise 10

The word house has 5 alphabets. If each new word should have 3 alphabets, then we should use the following formula:

\binom{n}{k} = \frac {n!} {k! (n-k!)}, where n \geq k \geq 0

Substitute the values in this example in the above formula:

\binom{5}{3} = \frac {5!} {3! (5-3!)}

\binom{5}{3} = 10

 

 

Solution of exercise 11

Number of pencils Sarah have = 8

Number of pencils she wants to give to her younger sister = 3

We will use the binomial coefficients formula to determine the number of combinations:

\binom{n}{k} = \frac {n!} {k! (n-k!)}, where n \geq k \geq 0

After substitution we will get the number of combinations:

\binom{8}{3} = \frac {8!} {3! (8-3!)}

\binom{8}{3} = \frac {8!} {3! (5!)}

\binom{8}{3} = \frac {40320} {720}

=56

Hence, Sarah can make 56 combinations of 8 colored pencils given the fact that she can choose 3 at a time.

 

Solution of exercise 12

Number of chocolates Alice have = 6

Number of chocolates she wants to give to her friend= 2

We will use the binomial coefficients formula to determine the number of combinations:

\binom{n}{k} = \frac {n!} {k! (n-k!)}, where n \geq k \geq 0

After substitution we will get the number of combinations:

\binom{6}{2} = \frac {6!} {2! (6-2!)}

\binom{6}{2} = \frac {6!} {2! (4!)}

\binom{6}{2} = \frac {720} {48}

=15

Hence, Alice can make 15 combinations of 6 chocolates given the fact that she can choose 2.

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.