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# A company plans to assign identification numbers to its

tagged by: AAPL

00:00

A

B

C

D

E

## Global Stats

Difficult

GMAT Prep

A company plans to assign identification numbers to its employees. Each number is to consist of four different digits from 0 to 9, inclusive, except that the first digit cannot be 0. How many different identification numbers are possible?

A. 3,024
B. 4,536
C. 5,040
D. 9,000
E. 10,000

OA B.

### GMAT/MBA Expert

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AAPL wrote:
GMAT Prep

A company plans to assign identification numbers to its employees. Each number is to consist of four different digits from 0 to 9, inclusive, except that the first digit cannot be 0. How many different identification numbers are possible?

A. 3,024
B. 4,536
C. 5,040
D. 9,000
E. 10,000

OA B.
Take the task of creating the identification numbers and break it into stages.

Stage 1: Select the first digit of the identification number
The first digit can be 1, 2, 3, 4, 5, 6, 7, 8, or 9
So, we can complete stage 1 in 9 ways

Stage 2: Select the second digit of the identification number
The second digit can be any digit from 0 to 9 OTHER THAN the digit chosen in stage 1
So, we can complete stage 2 in 9 ways .

Stage 3: Select the third digit of the identification number
The third digit can be any digit from 0 to 9 OTHER THAN the 2 digits digit chosen in stages 1 and 2
So we can complete this stage in 8 ways.

Stage 4: Select the fourth digit of the identification number
The third digit can be any digit from 0 to 9 OTHER THAN the 3 digits digit chosen earlier
So we can complete this stage in 7 ways.

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus create all of the identification numbers) in (9)(9)(8)(7) ways (= 4,536 ways)

--------------------------

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/video/775

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

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/counting-problem-company-recruitment-t244302.html
- https://www.beatthegmat.com/picking-a-5-digit-code-with-an-odd-middle-digit-t273110.html
- https://www.beatthegmat.com/permutation-combination-simple-one-t257412.html
- https://www.beatthegmat.com/simple-one-t270061.html

MEDIUM
- https://www.beatthegmat.com/combinatorics-solution-explanation-t273194.html
- https://www.beatthegmat.com/arabian-horses-good-one-t150703.html
- https://www.beatthegmat.com/sub-sets-probability-t273337.html
- https://www.beatthegmat.com/combinatorics-problem-t273180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-separator-method-t271047.html
- https://www.beatthegmat.com/combinatorics-problem-t267079.html

DIFFICULT
- https://www.beatthegmat.com/wonderful-p-c-ques-t271001.html
- https://www.beatthegmat.com/permutation-and-combination-t273915.html
- https://www.beatthegmat.com/permutation-t122873.html
- https://www.beatthegmat.com/combinations-t123249.html

Cheers,
Brent

_________________
Brent Hanneson – Creator of GMATPrepNow.com
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### GMAT/MBA Expert

GMAT Instructor
Joined
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Posted:
520 messages
Followed by:
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AAPL wrote:
GMAT Prep

A company plans to assign identification numbers to its employees. Each number is to consist of four different digits from 0 to 9, inclusive, except that the first digit cannot be 0. How many different identification numbers are possible?

A. 3,024
B. 4,536
C. 5,040
D. 9,000
E. 10,000
Immediate application of the Multiplicative Principle:

$\begin{array}{*{20}{c}} {\underline {{\text{not}}\,\,0} } \\ 9 \end{array}\begin{array}{*{20}{c}} {\underline {{\text{nr}}} } \\ 9 \end{array}\begin{array}{*{20}{c}} {\underline {{\text{nr}}} } \\ 8 \end{array}\begin{array}{*{20}{c}} {\underline {{\text{nr}}} } \\ 7 \end{array}\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{Multipl}}{\text{.}}\,{\text{Principle}}} \,\,\,\,? = {9^2} \cdot 8 \cdot 7\,\,\,\,\,\,\,\,\,\,\left[ {nr = {\text{no}}\,\,{\text{repetition}}} \right]$
$\left\langle ? \right\rangle = \left\langle {{9^2}} \right\rangle \cdot \left\langle {8 \cdot 7} \right\rangle = 1 \cdot 6 = 6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\left\langle N \right\rangle = {\text{units}}\,\,{\text{digit}}\,\,{\text{of}}\,\,N} \right]$

Just one alternative choice with unit´s digit equal to the correct one... we are done!

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: www.GMATH.net (Math for the GMAT)
Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!

### GMAT/MBA Expert

GMAT Instructor
Joined
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Posted:
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Followed by:
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AAPL wrote:
GMAT Prep

A company plans to assign identification numbers to its employees. Each number is to consist of four different digits from 0 to 9, inclusive, except that the first digit cannot be 0. How many different identification numbers are possible?

A. 3,024
B. 4,536
C. 5,040
D. 9,000
E. 10,000
There are 9 choices for the first digit (1 through 9, inclusive). The second digit can be any of the 10 digits (0 through 9, inclusive) EXCEPT it can’t repeat the first digit; thus, there are 9 options for the second digit. The third digit can’t repeat either of the first two digits, so there are 8 options. Similarly, the fourth digit can’t repeat any of the first 3 digits, so there are 7 options. Thus, the total number of options is 9 x 9 x 8 x 7 = 4,536.

_________________
Scott Woodbury-Stewart Founder and CEO

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