Question

The largest 2-digit number is 99. How many 2-digit numbers must be in a set in order to apply the pigeonhole principle to conclude that there are two distinct subsets of the numbers whose elements sum to the same value

Answer 1

Answer:

The solution can be defined as follows:

Step-by-step explanation:

$\to n = 10$

There are 210 = 1,024 sub-sets of 10 entries, however, the number of enters between the lower and upper limits total of 10 different submissions between 1 and 100 can be as low only as 901 (= 955 - 55 + 1). There has to be a minimum level of at least 1 subset with much more subgroups than possible, which equals at least two sets. Call for two main sub, S' and T' equivalent.

Let

$C = S' \cap T'$

when

$S = S' - C \\\\T = T' - C$

Then$S \ and\ T$ are disjoint

$sum(S) = sum(S' - C)$

            $= sum(S') - sum(C)\\\\= sum(T') - sum(C)\\\\= sum(T' - C)\\\\= sum(T)$