Alice makes 2-digit codes using the 26 letters of the alphab

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed Apr 18, 2018 3:48 am

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[GMAT math practice question]

Alice makes 2-digit codes using the 26 letters of the alphabet. Letters may be used repeatedly, and at least one vowel must be used. How many possible codes can she make?

A. 235
B. 256
C. 360
D. 625
E. 676

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Source: Beat The GMAT — Problem Solving | Wed Apr 18, 2018 3:48 am

Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Fri Apr 20, 2018 12:42 am

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The number of vowels is 5 (a,e,i,o,u). So, the number of codes is equal to the number of 2-digit codes with no restrictions minus the number of codes that contain no vowels. This is given by

26*26 – 21*21 = (26 + 21)(26 – 21) = 47*5 = 235.

Therefore, the answer is A.
Answer: A

Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Mon Apr 23, 2018 4:53 pm

Max@Math Revolution wrote:[GMAT math practice question]

Alice makes 2-digit codes using the 26 letters of the alphabet. Letters may be used repeatedly, and at least one vowel must be used. How many possible codes can she make?

A. 235
B. 256
C. 360
D. 625
E. 676

Each code must have at least one vowel. Thus, we have 3 cases for creating the codes: 1) vowel-consonant, 2) consonant-vowel, and 3) vowel-vowel.

Case 1: vowel-consonant

We have 5 x 21 = 105 possible codes.

Case 2: consonant-vowel

We also have 21 x 5 = 105 possible codes.

Case 3: vowel-vowel

We have 5 x 5 = 25 possible codes.

Thus, the total number of possible codes is 105 + 105 + 25 = 235.

Answer: A

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