Any Seven Will Do?

Probability Level 2

The sum of $n$ positive integers , not necessarily distinct, is 100. The sum of any 7 of them is less than 15. What is the minimum value of $n$ ?

The answer is 50.

1 solution

Lim Li Yen
Jul 26, 2016

Relevant wiki: Extremal Principle

Assume n=50 by taking 50 copies of 2. Then the sum of any 7 of them will be 14. Suppose n is less than 50. Divide these numbers into at most 7 groups, their sum will not more than 14. The sum of each group is at most 14 and the sum of all numbers is at most $7\times 14=98 \lt 100$ .

???......can you give a better explanation

abhishek kulkarni - 4 years, 10 months ago

If it there are 49 nummers at least one of them is larger then 2. So either there are 47 2's and 2 3's or 48 2's and a 4. If you take 7 numbers and the 3 or 4 is one of those the sum of those numbers is 15 or greater. But it most be max 14 so n=49 or less doesn't work.

Peter van der Linden - 4 years, 10 months ago

????? More explanation please

Jay Lim - 4 years, 10 months ago

And why does this solution exclude n>50?

Peter van der Linden - 4 years, 10 months ago

Forget that: MINIMUM is to be found...

Peter van der Linden - 4 years, 10 months ago

I had a doubt, why can't n be 100 ? 100 copies of 1 ? Since you had mentioned all numbers needn't be distinct and that sum of any 7 should be less than 15, then why doesn't n = 100 qualify ?

Arnav Das - 4 years, 10 months ago

Any Seven Will Do?

The answer is 50.

1 solution

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