Discrete Math -- Sets I am struggling with thinking about this. Any help would be great!!

A medical research survey categorizes adults as follows:


by gender (male or female)
by age group (age groups are 18-25, 26-35, 36-50, 51+)
by income (less than 30k/year, 30k-60k/year, more than 60k/year)
for women only: by whether they have been pregnant (yes/no)
for men only: by frequency of undergoing prostate exams (frequently, rarely, never).


What minimum size of a set of adults will guarantee that there are two people in it with matching characteristics in all categories? You do not need to explain your answer.
Based on this



I believe you would have (choices)


(5)

Woman : Yes

Woman : No

Male  : Frequently

Male  : Rarely

Male  : Never
(4) by age group (age groups are 18-25, 26-35, 36-50, 51+)
(3) by income (less than 30k/year, 30k-60k/year, more than 60k/year)


$$5*4*3=60$$

By the product rule, there are 60 ways of answering this survey. Therefore if,

$$60+1=61$$

61 people were surveyed, two will have matching characteristics.
By the Pigeonhole Principle it would be one more than the number of categories, C(5,2)*4*3 = 10*4*3 = 120 => 120 +1 = 121.