Combinations (order doesn’t matter) and repetition is not allowed.
(nk)=(85)=56
title: Correct answer.
We know first 2 slots are taken by, so check for last 3.
(63)=20
title: Correct answer.
Pick 5 from 6 as before.
(65)=6
title: Correct answer.
Consider where Mary or Peter are definitely there and the other is not in the pool (4 from 6).
2×(64)Total=2×15=30=20 (where both from before)+30=50
title: Correct answer.
This is the first part of above.
2×(64)=2×15=30
title: Correct answer.
2
In this case, we find permutations where repetitions are allowed.
It can either be heads or tails, 2 outcomes.
We flip it 8 times.
28=256
title: Correct answer.
If there are exactly three heads, repetition is not allowed and order matters doesn’t matter.
We have n=8 items which we are selecting from a total of k=3.
(83)=56
title: Correct answer.
How many ways can we choose 3 numbers from 8?
Assume we have already had 3 heads, find permutations where repetitions are allowed where n=2 and k=5.
25=32
title: Incorrect answer.This is actually all outcomes excluding those containing either none, one or two heads:$$ 2^8 - \begin{pmatrix} 8\\0 \end{pmatrix} - \begin{pmatrix} 8\\1 \end{pmatrix} - \begin{pmatrix} 8\\2 \end{pmatrix} = 219$$