Calculate the possibilities to form an exercise group

[rootmeister64]

In one lecture, there are 20 students S1, S2, ..., S20. How many possibilities are there to form an exercise group (of any size) that includes at least one of the following students: S1, S2, S3?

I came up with the following solution:

We have 20 students, and each student can be in a group or not (1 or 0).
Each group must have at least one student from this set {S1, S2, S3}. Therefore we have (2³ - 1) possibilities for the first three students ( -1 because there is also the possibility that none of the students are in the group {0, 0, 0}). The remaining students don't matter, thus we have 2¹⁷ possibilities.

Therefore we can build (2³ - 1) * 2¹⁷ groups.

Is the solution that came up with correct? Are there other ways to solve this problem?

Thank you very much!

 

[Dr.Peterson]

In one lecture, there are 20 students S1, S2, ..., S20. How many possibilities are there to form an exercise group (of any size) that includes at least one of the following students: S1, S2, S3?

I came up with the following solution:

We have 20 students, and each student can be in a group or not (1 or 0).
Each group must have at least one student from this set {S1, S2, S3}. Therefore we have (2³ - 1) possibilities for the first three students ( -1 because there is also the possibility that none of the students are in the group {0, 0, 0}). The remaining students don't matter, thus we have 2¹⁷ possibilities.

Therefore we can build (2³ - 1) * 2¹⁷ groups.

Is the solution that came up with correct? Are there other ways to solve this problem?

Sounds good.

Another way would be to subtract all the ways to form a group that does not include any of 1, 2, and 3, from all the ways to form a group: 2^20 - 2 ^17. This is the same answer.

 

[rootmeister64]

Sounds good.

Another way would be to subtract all the ways to form a group that does not include any of 1, 2, and 3, from all the ways to form a group: 2^20 - 2 ^17. This is the same answer.

Thank you!

 

[Steven G]

Lets make you a bit tougher. Suppose there were only 6 students in the class and you had the same constraint that you must choose at least one of the 1st three. Can you list all the possibilities and see if your formula works? Whenever someone asks me if they counted correctly (and the correct answer is not too large) I tell them that I do not know and suggest that they list all the possibilities and see if it matches their answer.

 

[rootmeister64]

Lets make you a bit tougher. Suppose there were only 6 students in the class and you had the same constraint that you must choose at least one of the 1st three. Can you list all the possibilities and see if your formula works? Whenever someone asks me if they counted correctly (and the correct answer is not too large) I tell them that I do not know and suggest that they list all the possibilities and see if it matches their answer.

I came up with the following idea to count the students:

What do you think about it?

Thank you!

 

[Steven G]

I came up with the following idea to count the students:
View attachment 25584
What do you think about it?

Thank you!

Now do you believe the formula which you used in your larger problem??

 

[rootmeister64]

Now do you believe the formula which you used in your larger problem??

Yes I consider the formula to be correct.