A company plans to assign identification numbers to its

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AAPL: Moderator; Posts: 2632; Joined: Sun Oct 29, 2017 2:08 pmFollowed by: 2 members

by AAPL » Wed Sep 12, 2018 5:35 am

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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.

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Source: Beat The GMAT — Problem Solving | Wed Sep 12, 2018 5:35 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

A company plans to assign identification numbers to its

by Brent@GMATPrepNow » Wed Sep 12, 2018 5:54 am

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)

Answer: B
--------------------------

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- https://www.beatthegmat.com/digits-numbers-t270127.html
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DIFFICULT
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Brent Hanneson - Creator of GMATPrepNow.com

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Wed Sep 12, 2018 1:38 pm

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 unitÂ´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 :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8136; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Fri Sep 14, 2018 8:25 am

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.

Answer: B

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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