Combinatorics

9000139308

Level:

The shooting club has

25

members. Among the members it is necessary to vote a board: a president, a cashier and a webmaster. One person cannot have more than one of these positions and there is only one member skilled enough to be a webmaster. How many possibilities exist to set up the board?

24 \cdot 23 = 552

25 \cdot 24 = 600

24 \cdot 23 \cdot 22 = 12 144

25 \cdot 24 \cdot 23 = 13 800

9000139309

Level:

There are

20

tablets in an e-shop. From this amount

18

tablets are new and

2

tablets have been returned by customers. The e-shop manager gets an order containing three tablets and he wants to get rid of the returned tablets first. How many possibilities exist to complete the order?

18

\frac{18!}{3! 15!} = 816

18 \cdot 16 \cdot 3 = 864

20 \cdot 19 \cdot 18 = 6 840

9000139310

Level:

There are

20

tablets in an e-shop. From this amount

18

tablets are new and

2

tablets have been returned by customers. The e-shop manager gets an order containing three tablets and he wants to use only the new tablets for this order. How many possibilities exist to complete the order?

\frac{18!}{3! 15!}

18

18 \cdot 16 \cdot 3

20 \cdot 19 \cdot 18

9000139701

Level:

There are

15

athletes in an athletic meeting. Determine in how many ways it is possible to obtain the results on the first six places of the scoreboard if the place on scoreboard cannot be shared (one athlete per one place on scoreboard).

\frac{15!}{9!} = 3 603 600

6^{15} = 470 184 984 576

15! 6! = 941 525 544 960 000

\frac{15!}{9! 6!} = 5 005

9000139702

Level:

A race was attended by

12

competitors. Find the number of possible orderings of the competitors on the first three places of the scoreboard.

\frac{12!}{9!} = 1 320

3^{12} = 531 441

\frac{12!}{9! 3!} = 220

12! 3! = 2 874 009 600

9000139703

Level:

The box contains

5

red crayons,

4

yellow crayons and

2

green crayons. The crayons are removed from the box and arranged in a line. How many different color patterns can be obtained by this procedure?

\frac{11!}{5! 4! 2!} = 6 930

5 \cdot 4 \cdot 2 = 40

5! 4! 2! = 5 760

{(5! 4!)}^{2} = 8 294 400

9000139705

Level:

From the group of

10

boys and

5

girls we have to select a small group of

3

boys and

2

girls. How many possibilities exist for this choice?

\frac{10!}{7! 3!} \cdot \frac{5!}{3! 2!} = 1 200

5^{10} = 9 765 625

10 \cdot 5! 3! = 7 200

5 \cdot \frac{10!}{3!} = 3 024 000

9000139706

Level:

The international alphabet contains

26

letters. The letters of this alphabet and the digits from

0

9

are used to form a code of the length

4

(a code contains

4

characters). The characters may repeat through the code and the code is not case sensitive (uppercase letters are equivalent to lowercase letters). How many codes can be obtained?

36^{4} = 1 679 616

10 \cdot 26^{4} = 4 569 760

\frac{36!}{32! 4!} = 58 905

\frac{26!}{22! 4!} = 14 950

9000139707

Level:

A Morse code utilized dots and dashes to encode letters of an alphabet. Find the number of signals of the length from

1

4

which can be obtained from dots and dashes.

2 + 2^{2} + 2^{3} + 2^{4} = 30

1 + 2 + 3! + 4! = 33

\frac{4!}{3! 2!} = 2

2 \cdot 1 + 2 \cdot 2 + 2 \cdot 3 + 2 \cdot 4 = 20

9000139708

Level:

The shelf contains

15

books. From this amount,

9

books are in English and

6

books in other languages. Find the number of possibilities how to rearrange the books on the shelf, if all English books have to be on the left and the other on the right.

9! 6! = 261 273 600

9^{6} = 531 441

\frac{9!}{6!} = 504

\frac{9!}{6! 3!} = 84

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