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A positive 5-digit integer is called "friendly" if

by fskilnik@GMATH » Tue Feb 12, 2019 7:05 am

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GMATH practice exercise (Quant Class 18)

A positive 5-digit integer is called "friendly" if each of its digits is 1,2,3,4 or 5 and all of its digits are different. Putting all friendly numbers in increasing order, the units digit of the friendly number that occupies the 32nd place is:

(A) 5
(B) 4
(C) 3
(D) 2
(E) 1

Answer: [spoiler]____(B)__[/spoiler]
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by GMATGuruNY » Tue Feb 12, 2019 7:41 am
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 18)

A positive 5-digit integer is called "friendly" if each of its digits is 1,2,3,4 or 5 and all of its digits are different. Putting all friendly numbers in increasing order, the units digit of the friendly number that occupies the 32nd place is:

(A) 5
(B) 4
(C) 3
(D) 2
(E) 1


Total number of friendly numbers = total number of ways to arrange the 5 digits = 5! = 120.
There are 5 options for the first digit: 1, 2, 3, 4, or 5.
120/5 = 24.
Thus, the 120 friendly numbers are composed of five 24-number ranges, as follows:
The integers beginning with 1 constitute friendly numbers 1-24.
The integers beginning with 2 constitute friendly numbers 25-48.
The integers beginning with 3 constitute friendly numbers 49-72.
The integers beginning with 4 constitute friendly numbers 73-96.
The integers beginning with 5 constitute friendly numbers 97-120.

The 32nd friendly number is contained within the blue range above.

Friendly numbers in the blue range, from least to greatest:
21345 --> 25th friendly number
21354 --> 26th friendly number
21435 --> 27th friendly number
21453 --> 28th friendly number
21534 --> 29th friendly number
21543 --> 30th friendly number
23145 --> 31st friendly number
23154 --> 32nd friendly number

The units digit for the 32nd friendly number = 4.

The correct answer is B.
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by fskilnik@GMATH » Tue Feb 12, 2019 5:49 pm
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 18)

A positive 5-digit integer is called "friendly" if each of its digits is 1,2,3,4 or 5 and all of its digits are different. Putting all friendly numbers in increasing order, the units digit of the friendly number that occupies the 32nd place is:

(A) 5
(B) 4
(C) 3
(D) 2
(E) 1


FOCUS: the units digit of the 32nd friendly number

1. The first P4 = 4! = 24 friendly numbers "start with 1" (i.e., 1 _ _ _ _ ) , the (few) next ones "start with 2".

2. "Starting with 2" (i.e., 2 _ _ _ _ ), there are P3 = 3! = 6 friendly numbers "beginning with 21" (i.e., 2 1 _ _ _ ), hence:
All friendly numbers, from the 25th until the 30th (both included) are of the form 21 _ _ _ .

3. The 31st friendly number is the smallest friendly number that starts with "23", that is, 23145.

4. Finally, the 32nd friendly number is the next one (23154), hence ? = 4.


The correct answer is (B).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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