Problem Solving

BTGModeratorVI wrote: ↑

Mon Apr 13, 2020 3:39 pm

A men's basketball league assigns every player a two-digit number for the back of his jersey. If the league uses only the digits 1-5, what is the maximum number of players that can join the league such that no player has a number with a repeated digit (e.g. 22), and no two players have the same number?

A. 20
B. 21
C. 22
D. 24
E. 25

Answer: A
Source: Manhattan prep

Take the task of creating different jersey numbers and break it into stages.

Stage 1: Select the first digit
There are 5 digits (1, 2, 3, 4, or 5) to choose from, so we can complete stage 1 in 5 ways

Stage 2: Select the second digit
Repeated digits are NOT ALLOWED.
So, once we select the 1st digit in stage 1, we cannot select it again.
So, there are 4 digits remaining to choose from.
We can complete stage 2 in 4 ways

By the Fundamental Counting Principle (FCP), we can complete the two stages (and thus create the 2-digit jersey numbers) in (5)(4) ways ( = 20 ways)

Answer: A

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch this free video: https://www.gmatprepnow.com/module/gmat- ... /video/775

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat ... /video/776

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/permutation-t122873.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/combinations-t123249.html


Cheers,
Brent

BTGModeratorVI wrote: ↑

Mon Apr 13, 2020 3:39 pm

A men's basketball league assigns every player a two-digit number for the back of his jersey. If the league uses only the digits 1-5, what is the maximum number of players that can join the league such that no player has a number with a repeated digit (e.g. 22), and no two players have the same number?

A. 20
B. 21
C. 22
D. 24
E. 25

Answer: A
Source: Manhattan prep

Solution:

We need to determine how many two-digit numbers can be created from 5 digits (1 to 5, inclusive), with no repeated digits. Since order matters, we have a permutation. Thus, the number of ways to create two-digit numbers is 5P2 = 5!/(5 - 2)! = 5 x 4 = 20.

Alternate Solution:

If repeated digits were allowed, there would be 25 possibilities since, for each digit, we would have 5 choices. Among these 25 possibilities, 5 of them are repeated digit numbers (which are 11, 22, 33, 44, and 55). Thus, without the repeated digits, there are 25 - 5 = 20 numbers possible.

Answer: A