If the first digit cannot be a 0 or a 5, how many five-digit odd numbers are there?
A. 42,500
B. 37,500
C. 45,000
D. 40,000
E. 50,000
Soln: This problem can be solved with the Multiplication Principle. The Multiplication Principle tells us that the number of ways independent events can occur together can be determined by multiplying together the number of possible outcomes for each event.
There are 8 possibilities for the first digit (1, 2, 3, 4, 6, 7, 8, 9).
There are 10 possibilities for the second digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
There are 10 possibilities for the third digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
There are 10 possibilities for the fourth digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
There are 5 possibilities for the fifth digit (1, 3, 5, 7, 9)
Using the Multiplication Principle:
= 8 * 10 * 10 * 10 * 5
= 40,000
(D).
I understand the solution and problem. However, I guess I don't completely understand it because I used 8!10!10!10!5! instead of the Multiplication Principle stated above.
My reasoning was that there are x! number of ways to arrange each possible digit.
Please guide! Thank you very much.
Factorials versus Multiplication
This topic has expert replies
-
- Master | Next Rank: 500 Posts
- Posts: 163
- Joined: Fri Sep 26, 2008 10:54 pm
- Thanked: 7 times
If you are given 8 digits and asked to form all possible 8 digit numbers. Then you say 8! ways.I used 8!10!10!10!5! instead of the Multiplication Principle stated above.
BUT
If you are given 8 digits and asked to pick any one out of it. Then you have only 8 ways . You are given five such groups and asked to frame a 5 digit number. so you multiply the length of all groups.
Hope you are clear
- Stuart@KaplanGMAT
- GMAT Instructor
- Posts: 3225
- Joined: Tue Jan 08, 2008 2:40 pm
- Location: Toronto
- Thanked: 1710 times
- Followed by:614 members
- GMAT Score:800
Exactly - when looking at a string of numbers, we'd use permutations. When looking at each individual digit, it becomes a combinations issue.jeevan.Gk wrote:If you are given 8 digits and asked to form all possible 8 digit numbers. Then you say 8! ways.I used 8!10!10!10!5! instead of the Multiplication Principle stated above.
BUT
If you are given 8 digits and asked to pick any one out of it. Then you have only 8 ways . You are given five such groups and asked to frame a 5 digit number. so you multiply the length of all groups.
Hope you are clear
Here's another way you could think about it:
1st digit.. pool of 8 numbers, choosing 1... 8C1
2nd-4th digits... pool of 10 numbers, choosing 1.. 10C1 (each)
last digit... pool of 5 digits, choosing 1... 5C1
So, we're multiplying:
8C1 * 10C1 * 10C1 * 10C1 * 5C1 = 8*10*10*10*5
which is really just the multiplication principle explained a different way (that can be applied to more complicated questions as well, i.e. when we're choosing more than 1 member from each group).
Stuart Kovinsky | Kaplan GMAT Faculty | Toronto
Kaplan Exclusive: The Official Test Day Experience | Ready to Take a Free Practice Test? | Kaplan/Beat the GMAT Member Discount
BTG100 for $100 off a full course