Question

Nine stuffed animals, all different, are placed along a babies toy shelf. How many possible

arrangements are there if:

1. there are no restrictions.

2. Cookie Monster must be in the middle.

3. BERT and ERNIE must be together.

4. BERT, ERNIE, and OSCAR must be together.

5. BERT, ERNIE, and OSCAR must not be together.

6. they must not be in alphabetical order.

Answer 1

a) Given there are no restrictions, number of ways to arrange 9 toys is computed as the permutation of 9 items

= 9!

Therefore there are 9! = 362880 ways to arrange the toys here.

b) The number of arrangements possible here such that Cookie monster is in the middle is computed as:

= 8! since we already have a position for one of them

Therefore there are 8! = 40320 ways to arrange the toys here.

c) The number of ways to arrange such that BERT and ERNIE must be together is computed here as:

= Number of permutation of 8 items where one item is BERT and EARNIE together * Number of permutation of BERT and EARNIE

= 8!*2

= 80640

therefore there are 80640 ways to arrange here.

d) Similar to above part, here we would be left with 7 total items but permutation of 3 items within themselves would be 3!.

Therefore total number of arrangements:
= 7! * 3!

= 30240

therefore there are 30240 ways to arrange here.

e) Arrange the remaining 5 toys first in 5 ! ways and then we ahve 6 spots to place the three toys which cant be together. Therefore the number of ways to arrange the remaining 3 would be permutation of 6 items taken 3 at a time

= 5! * 6*5*4

= 14400

Therefore there are 14400 ways to arrange here.

f) Given that they must not be in alphabetical order, which can only happen in 1 way, therefore total number of arrangements possible here is computed as:

= 9! - 1

= 362879

therefore there are 362879 ways to arrange here.