In a certain nation, every citizen is assigned an identification numbe

Question

In a certain nation, every citizen is assigned an identification number consisting of the last two digits of the person’s birth year, followed by five other numerical digits. For instance, a person born in 1963 could have the identification number 6344409. How many identification numbers are possible for people born in the years 1980–1982, inclusive?

(A) 360
(B) 2,880
(C) 288,800
(D) 300,000
(E) 2,400,000

Show Answer

Official Answer

D

(A) 360
(B) 2,880
(C) 288,800
(D) 300,000
(E) 2,400,000

QuantMadeEasy · Answer

Step 1: Understanding the question 
Identification number consists of last two digits of the person’s birth year, followed by five other numerical digits.
Using slot approach, __ x __ x __ x __ x __ x __ x __ , total seven slots to fill

First slot filled by digit 8 i.e. 1 way 
Second slot filled by 0 or 1 or 2 i.e. 3 ways 
All other slots filled by any digit from 0 to 9 i.e.10 ways

Step 2: Calculation 
1 x 3 x 10 x 10 x 10 x 10 x 10 = 300,000

D is correct

KetakiUpadhye · Answer

Hi Quantmadeeasy 
How can we assume 10 ways for 5 slots... They have clearly mentioned other numbers which would mean that we cant repeat the prev numbers. Please help

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IanStewart · Answer

We have 3 choices for the first two digits (80, 81, 82), and then 10 choices for each of the five remaining digits, for 3*10^5 choices in total, which clearly must be answer D.

I agree with you that the wording of the problem is bad -- there is no reason to include the word "other" in the question. They're trying to say that the id number includes five additional digits, after the initial 80, 81 or 82, and not that the digits are different from the ones already used. The example id number they provide doesn't actually clarify that issue (it illustrates that repetition is allowed within the final five digits, but not whether we can reuse either of the first two digits).

If you did think you could not reuse either of the first two digits (so if an id number started with 81, if you thought, completely understandably, that you could only use the eight other digits for the rest of the id number) then the answer becomes 3*8^5, which (using a calculator) is slightly less than 100,000, and there's no answer choice that size. But that's the only reason I'm sure I've interpreted the problem as they intended. Your interpretation is just as reasonable, from the way the problem is worded. You won't face this issue in official questions, which are always worded very carefully to avoid any ambiguity.

KetakiUpadhye · Answer

Thank you for the explanation IanStewart... It was very helpful 😁

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Vibhatu · Answer

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In a certain nation, every citizen is assigned an identification numbe

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