How many pairs of positive integers is the sum less than 14?

[brucejin]

How many pairs of positive integers is the sum less than 14?

My work:

choose 1, there are 12 pairs: (1,1), (1,2).... (1,12)
choose 2, there are 11 pairs: (2,1), (2,2).... (2,11)
...
choose 12, there are 1 pair: (12,1)

so there are total of
1 + 2 + 3 + ... + 12 = 78 pairs.

My book says there are 91 pairs (must be 1 + 2 + 3 + ... + 12 + 13)

Is my book wrong?
or me?

 

[mmm4444bot]

It's possible that the number 91 in your book is a typographical error.

There are 41 pairs of positive integers whose sum is 13 or less. (I'm not even sure for what the actual exercise asks because your problem statement is not grammatically correct, as typed.)

When we combine (for example) the first two positive integers, that's only one pair.

In other words "1 and 2" is the same pair (combination) as "2 and 1".

If the book's author intends to count these as two distinct pairs, then the author needs to explicitly state "ordered pairs" or "permutations".

And, if this exercise actually does count ordered pairs, then I see another possibility. If the information in the book is expressed as an inequality (versus an English sentence), then perhaps the inequality symbol is a typographical error.

There are 91 ordered pairs of positive integers whose sum is less than or equal to 14.

Something is clearly wrong somewhere; I do not have enough information to determine what it is.

 

[brucejin]

The question in the book (2001 mathcounts school manual) is

For How many pairs of positive integers is the sum less than 14?

I missed the word 'for' in my first post.