Two tests in one day

William said:

A student is taking 4 courses and there is to be a test in each course next week(Monday to Friday). What is the probability that there will be at least two tests on the same day?

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There are \(\displaystyle \dbinom{4+5-1}{4}\) ways the four tests can be given in five days.

How many ways can the tests all be given on different days?

Find the probability that they are given on different days. Then find the complement probability.

why is it 4+5-1?

William said:

A student is taking 4 courses and there is to be a test in each course next week(Monday to Friday). What is the probability that there will be at least two tests on the same day?

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William said:

why is it 4+5-1?

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This topic is known as counting multisets.

There is a simple model of your question.
Suppose there is a box with five labeled compartments (the days) and there are four identical red balls.
How many ways can the balls be placed in that box.

Use the multiset counting formula: \(\displaystyle n=5,~k=4,~\dbinom{4+5-1}{4}\).

There are five ways that the balls can be placed in four different compartments.
If we subtract that from the total, we have the number of ways that at least one compartment has more then one( at least two) balls in it.

Ok - we haven't done multisets - is there a way of doing the question with simple ordered counting techniques(permutations, combinations)? I new I would be doing the probability of all the tests on seperate days and then subtracting from one to get the probability of 2 or more in a day but I can't get the total number of ways that 4 tests can be written in 5 days, or how many ways each day only has one test without just wrting it out.

for every day of the week isnt there 5 possibilities? no test, 1 test, 2 tests,...., 4 tests? Would all the different ways be 5 ^4?

William said:

for every day of the week isnt there 5 possibilities? no test, 1 test, 2 tests,...., 4 tests? Would all the different ways be 5 ^4?

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\(\displaystyle 5^4\) is the number of functions from a set of four to a set of five.
In this case, it is clear that there are five different days of the school week.
How are you thinking about this question?

Look at the string of symbols: \(\displaystyle 0~0~0~0~|~|~|~|\)

If I rearrange that string as \(\displaystyle \underbrace {}_M|\underbrace o_T|\underbrace {}_W|\underbrace {oo}_{Th}|\underbrace o_F\). That means one test on Tues., two on Thru. and one on Fri.

So away we can rearrange string represents a way to give four test in five days: \(\displaystyle \dfrac{8!}{(4!)^2}\).

You tell us how else you can count the ways.

Ok that I get

just by drawing out the ways there are 5 ways that at most only one test a day - is there a way to show that mathematically?

Actually the breaks are identical but the tests are not - so shouldn't it be 8!/4!? and then there would be more than 5 ways of having at most 1 test a day?

William said:

Actually the breaks are identical but the tests are not - so shouldn't it be 8!/4!? and then there would be more than 5 ways of having at most 1 test a day?

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In this the tests are identical objects because we are not concerned with individual tests but with the number of tests given on any day not which test.