Question

1. You own 6 songs by Adele, 4 by Katy Perry, and 5 by Lady
Gaga. How many different playlists can you make that consist of 4
Adele songs, 3 Perry songs, and 2 Gaga songs, **if you do
allow repeated songs?**

6^{4}*4^{3}*5^{2}=2073600 is not the
right answer.

2. How many permutations of {A, B, C, D, E, F, G, H, I} are there in which the first character is A, B, or C and the last character is G, H, or I ?

Example: B G D F C A E I H.

3. A bowl of fruit contains apples, bananas, oranges, and pears (at least 15 of each kind).

(a) In how many ways can we choose 15 pieces?

(b) In how many ways can we choose 15 pieces, including at least 5 oranges?

(c) In how many ways can we choose 15 pieces so that there are at most 4 bananas?

I tried C(60,15) for 3.a, C(15, 5) * C(55, 10) for 3.b, and C(45,15) + C(45, 4)*15 for 3.c, none worked.

This is for a discrete structures class, thank you.

Answer #1

#1.

4 Adele songs can be selected in 6C4 ways

3 Perry songs can be selected in 4C3 ways

2 Gaga songs can be selected in 5C2 ways

In a playlist order of songs matter hence the 9 songs can be arranged in 9! different ways.

Hence possible number of playlists are 6C4 * 4C3 * 5C2 * 9! = 217728000

#2.

First character can be selected in 3C1 ways and last character can
be selected in 3C1 ways

Remaining 7 characters can be arranged in 7! ways

Hence total possible permutations are 3*3*7! = 45360

#3.

w - apples, x - bananas, y - oranges and z - pears

a)

w + x + y + z = 15

Number of possible ways are (15 + 4 - 1) C (4 - 1) = 18C3 = 816

b)

if we remove the 5 oranges from the above equation

w + x + y + z = 10

Number of possible ways are (10 + 4 - 1) C (4 - 1) = 13C3 = 286

c)

lets consider the combinations with at least 5 bananas, then
equation will be

w + x + y + z = 10

Number of possible ways are (10 + 4 - 1) C (4 - 1) = 13C3 = 286

Hence possible number of ways are 816 - 286 = 530

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