Answer to Question #218098 in Discrete Mathematics for Davie

Question

Basic Counting Principle

6.     How many different car license plates can be constructed if the licenses contain three letters followed by two digits if:

a.) Repetitions are allowed;

b.) repetitions are not allowed.

7.     Two dice are rolled, one blue and one red. How many outcomes have either the blue die 3 or an even sum or both?

8.     How many integers from 1 to 10,000, inclusive, are multiples of 5 or 7 or both?

9.     Prove that if five cards are chosen from an ordinary 52-card deck, at least two cards are of the same suit.

10. Eighteen persons have first names Adrian, Jheo and Ghimel and last names Ablir and Testor. Show that at least three persons have the same first and last names.

Answer 1

Solution.

6.a) repetitions are allowed

There are 26 letters (A-Z) and 10 digits (0-9). We can choose each of the first three positions in 26 ways.

Fourth and fifth positions can be chosen in 10 ways.

26•26•26•10•10=1757600

There are 1757600 different car license plates.

6.b) repetitions are not allowed

There are 26 letters (A-Z) and 10 digits (0-9). We can choose the first position in 26 ways, the second position in 25 ways, the third position in 24 ways, the fourth position in 10 ways, and the fifth position in 9 ways.

26•25•24•10•9=1404000.

There are 1404000 different car license plates.

7.

Use the Inclusion-Exclusion Principle to solve. Let be $A_1$ =blue die is 3, $A_2=$ even sum.

Then $|A_1\bigcup A_2|=|A_1|+|A_2|-|A_1\bigcap A_2|= 6+18-3=21.$

Thus there are 21 possible outcomes.

8.

For the number 5 there are 10000:5=2000 numbers divisible by 5 from 1 to 10000.

For the number 7 there are [10000:7]=1428 numbers divisible by 7 from 1 to 10000.

LCM(5,7)=35. Since for the number 35 there are [10000:35]=285 numbers divisible by 35 from 1 to 10000.

2000+1428-285=3146 integer numbers are multiples of 5 or 7 or both.

9.

The Pigeonhole Principle:

If n pigeons fly into k pigeonholes and k<n,

some pigeonhole contains at least two pigeons.

We have of the 5 cards as 5 pigeons and the 4

suits as 4 pigeonholes.

By the PHP, some suit (pigeonhole) is assigned to at least two cards (pigeons).

10.

Let the cards be the people and let the pigeonholes be the combinations of first and last name in The Pigeonhole Principle.

Thus 6 < 18, then at least three persons have the same first and last names.