Problem Solving

tata wrote: ↑

Mon Mar 01, 2010 10:30 am

A palindrome is a number that reads the same forward and backward, such as 121. How many odd, 4-digit numbers are palindromes?
A. 40
B. 45
C. 50
D. 90
E. 2500

Take the task of building palindromes and break it into stages.
Begin with the most restrictive stage.

Stage 1: Select the units digit
We can choose 1, 3, 5, 7 or 9
So, we can complete stage 1 in 5 ways

Stage 2: Select the tens digit
We can choose 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9
So, we can complete stage 2 in 10 ways

IMPORTANT: At this point, the remaining digits are already locked in.

Stage 4: Select the hundred digit
This digit must be the SAME as the tens digit (which we already chose in stage 2)
So, we can complete this stage in 1 way.

Stage 5: Select the thousands digit
This digit must be the SAME as the units digit (which we already chose in stage 1)
So, we can complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus build a 4-digit palindrome) in (5)(10)(1)(1) ways ([spoiler]= 50 ways[/spoiler])

Answer: C
--------------------------

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

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
- https://www.beatthegmat.com/mouse-pellets-t274303.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/ps-counting-t273659.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/please-solve ... 71499.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/laniera-s-co ... 15764.html

Cheers,
Brent

tata wrote: ↑

Mon Mar 01, 2010 10:30 am

A palindrome is a number that reads the same forward and backward, such as 121. How many odd, 4-digit numbers are palindromes?
A. 40
B. 45
C. 50
D. 90
E. 2500




[spoiler]OA = C [/spoiler]


I tried to search this topic but couldnt find it. The source is MGMAT, I couldnt get their explanation. Please give your inputs with detailed explanations.

Thanks in advance.

So we have the 4-digit numbers in the form of ABBA where A is an odd number and B can be any digit including B = A.

Therefore, we have 5 choices for the first A and 10 choices for the first B. However, since the second A and B must be the same as the first A and B, respectively, there is only 1 choice for each of the second A and B. So we have 5 x 10 x 1 x 1 = 50 such numbers.

Answer: C