# How do I approach this combinatorics problem about loan applications?

When acting on loan applications it can be concluded, based on historical records, that loan applicants having certain combinations of features can be expected to repay their loans and those who have other combinations of features cannot. As their main features, suppose that a bank uses:

Marital Status: Married, Single (never married), Single (previously married).

Past Loan: Previous default, No Previous default

Employment: Employed, Unemployed (within 1 year), Unemployed (more than 1 year).

(a) How many different loan applications are possible when considering these features?

(b) How many manifestations of loan repayment/default are possible when considering these features?

For part a, it seems plain to me that it's the product rule and is 3 * 2 * 3 = 18. But I'm confused about the wording in (b) - what are manifestations , and how do I approach this part? I appreciate any tips or advice.

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I think they mean that there are $2^{18}$ ways of drawing up the truth table of marital status / past loan / employment vs whether the load will be repaid or defaulted. –  user22805 Feb 29 '12 at 10:02
oops You are right, Thank You Very Much. Hmm, I'll think more! –  Adel Feb 29 '12 at 10:03
@David: Just $2\cdot 18=36$, two for each of the $18$ possible application types. –  Brian M. Scott Feb 29 '12 at 10:08
@Adel: If @David’s interpretation is right, you’re just adding a fourth category, Repay/Default, with two possible outcomes, so you can use the same kind of reasoning that you used to get $3\cdot 2\cdot 3$ in (a). –  Brian M. Scott Feb 29 '12 at 10:09
@Brian - no worries. I just hope that I haven't confused the original poster too much with my misinterpretation and our subsequent discussion. –  user22805 Mar 1 '12 at 0:26

## 1 Answer

It seems to me that the reasonable interpretation is that for every one of the $18$ classes that a particular person can fall into, there are $2$ possibilities for the actual outcome (default or not default), for a total of $36$.

What I would add is that the usage of the word "manifestation" in the question has a definite French tinge. Although using "manifestation" (pronounce it in English) here is not quite idiomatic, if you pronounce it in French, it is perfectly idiomatic.

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Thank You Very Much! This makes a lot of sense now! –  Adel Mar 3 '12 at 0:11