Show that if you have 15 employees, then at least 2 will have a birthday in the

Question

Show that if you have 15 employees, then at least 2 will have a birthday in the same month. You pay your employees every 2 weeks. Show that some months they will get 3 checks. Students have choice of 4 english classes (ENG 101, 102, 205, 300), and 2 Math classes (MATH 100, MATH 400). Students have to choose 1 English and 1 math class. How many students do you have to have before you know that 2 students share the same combination of classes? How many students do you have to have before you know that 4 students share the same combination of classes?

Explanation / Answer

13. We know that there are 12 months in a year.Here we have 15 employees.

Since we are trying to find the worst case of atleast 2 employees fall in same month.

Best way of distributing the12 employees among 12 months is 1 employee per month

And the remaining 3 employees can be distributed among 12 months like

CASE1:

3 in one month ,i.e 3+1=4 in one month(Here 1 refers to the 12 employees we have arranged among 12 months initially)

CASE2:

2 in one month 1 in other i.e,2+1=3 in one month 1+1=2 in another

CASE3:

1 in one month 1 in second month and 1 in third month,i.e,1+1=2 in one month, 1+1=2 in second month, 1+1=2 in third month

Therefore, considering all the cases we observe that there are a minimum of 2 will have birthday in same month.Hence proved.

14.There are about 52 weeks in a year and we pay employee every 2 weeks.

Dividing 52 weeks/ 2 weeks =26 times we pay employee in a year.

Every month has a minimum of 28 days (including Februaury) and hence there are atleast 4 weeks in a month(4*7days in a week=28 days).Hence you pay atleast 2 times a month.

For 12 months you pay 12*2 minimum 24 times , hence the remaining 2 times(26-24=2) should fall any two months(2+1=3 times in any two months)

Hence atleast few months employee gets paid 3 times, hence proved.

15.There are 4 english classes and 2 math classes and student has to choose 1 english and 1 math class.

Number of possible combinatioms in which a student can select 1 english and 1 math class(here we are multiplying both the cases because both choosing math and english class has to occur simultaneously)=(Selecting one english class out of 4) * (Selecting one math class out of 2)=4C1*2C1=4*2=8 different cominations are possible.

(a)For atleast 2 students to share same combinations,Since there are 8 combinations,if we choose 8 people then they could choose different combinatiins but if we choose atleast 9 ,we observe that even if we have 8 students choosing all different subject , the nineth student has to select one out of 8 combinations , then atleast 2 share same combinations.Therefore answer is 9 students.

(b)Similarly for atleast 4 students to share same combinations of classes.Suppose 8 students select 8 different subjects , then we need to have atleast 3 students selecting only 1 combination other than those 8 students.Then atleast 1 combination,will have 3+1(this 1 is from the 8 students distributed among 8 combinations)=4 if 8+3=11 students are presnt,hence anser is 11 students.

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