Question

A survey company asks all respondents for demographic information. There are 8 age and 7 income groups, 5 gender choices, and 6 choices for ethnicity. How many people would this company have to poll so it’s inevitable that at least three people in the sample have identical demographic information? Give the number and explain.

Answer 1

3361 people should poll so it's inevitable that at least three people in the sample have identical demographic information.

Explanation:

Consider the provided information.

There are 8 age and 7 income groups, 5 gender choices, and 6 choices for ethnicity.

For age we have 8 choices.

For Income groups we have 7 choices.

For gender groups we have 5 choices.

For ethnicity we have 5 choices.

Therefore, the total number of groups is: 8×7×5×6=1680

That means 1680 distinct group.

According to Pigeons holes principal: If m pigeons are in n holes and m > n, then at least 2 pigeons are in the same hole.

So to select at least 2 people in the sample have identical demographic information 1681 people needs to poll.

In order to at least 3 people (2 × 1680+1) people should need to poll.

2 × 1680 + 1 = 3361

Hence, 3361 people should poll so it's inevitable that at least three people in the sample have identical demographic information.