Two codes each are made of a combination of 6 digits or letters.

Question

Here's a quant question on combination/permutation I received this morning.

Two codes, called TR1 and TR2, each are made of 6 digits/letters.
TR1: First two are any digits in (0-9), last four are any letters in A-Z excluding AEIOU
TR2: First four are any digits in (0-9), last two are any letters in A-Z excluding AEIOUY
What's the ratio of possibilities of TR1 to possibilities of TR2 ?
(A-Z excluding AEIOUY has 20 characters)

Sorry for the bad writing, I just remember the gist of it.

EMPOWERgmatRichC · Answer

Hi kahipz,

I'm going to assume that the prompt is complete, but that there is a TYPO (we are probably meant to exclude AEIOUY from BOTH codes). The following number of codes are possible for TR1 and TR2:

For TR1, there are (10)(10)(20)(20)(20)(20) possible codes

For TR2, there are (10)(10)(10)(10)(20)(20) possible codes

The question asks for the ratio of TR1 'possibilities; to TR2 'possibilities'....

/

In this fraction, a number of the 'terms' will cancel out, leaving us with...

(1)(1)(2)(2)(1)(1) = 4/1

GMAT assassins aren't born, they're made,
Rich

kahipz · Answer

Thanks Rich!

I'm really bad at this combination/permutation quant topic and your answer renders it downright simple and easy.

Regarding the question, there's no typo. The last four letters of TR1 are made of letters from A-Z excluding vowels, which are A, E, O, U, I. The last two letters of TR2 are made of letters from A-Z excluding vowels AND letter Y, which are altogether A, E, O, U, I, Y.

But I suppose we need only a little tweak to solve the problem, as follows:

For TR1, there are (10)(10)(21)(21)(21)(21) possible codes

For TR2, there are (10)(10)(10)(10)(20)(20) possible codes

EMPOWERgmatRichC · Answer

Hi kahipz,

If there's no typo in the original prompt, then the correct answer to the question will look a lot 'uglier'...

As you've noted the possible number of codes would be...
For TR1, there are (10)(10)(21)(21)(21)(21) possible codes
For TR2, there are (10)(10)(10)(10)(20)(20) possible codes

So the ratio of TR1 codes to TR2 codes would be...

/

GMAT assassins aren't born, they're made,
Rich

kahipz · Answer

Hi Rich, That's true. The question may have asked for an estimate, or I may have not recalled the question correctly in its entirety. In any case, I do recall the gist of the problem and you've provided the core solution. Thank you for that :-D

May I ask you one more question on this quant topic using the problem at hand? Let's assume that we only need to calculate the number of possibilities of TR1, with new set of requirements: still 6 slots, each is made of either letters from A-Z or digits from 0-9, with at most 3 slots made of letters excluding vowels. This is a self-fabricated question and I'm unsure if it's sensical. If you think it's not, then ignore it; otherwise how'd you solve it

EMPOWERgmatRichC · Answer

Hi kahipz,

With the situation that you're describing, the math would get 'big', which is something that the GMAT writers don't do too often.

For the sake of the example, let's say that there are 20 letters and 10 digits.

For a six-character code to have AT MOST 3 letters, we would have to calculate the number of codes with....

3 letters and 3 digits
2 letters and 4 digits
1 letter and 5 digits
0 letters and 6 digits

OR

Subtract the number of codes with....

4 letters and 2 digits
5 letters and 1 digit
6 letters and 0 digits

...from the total number of possible codes.

Either way, we'd have to do a LOT of math and use the Combination Formula (to try to save some time on the sheer volume of work that would be involved).

GMAT assassins aren't born, they're made,
Rich

Two codes each are made of a combination of 6 digits or letters.

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