Math Multiplication Principle and Addition Principle

6. Suppose the first 4 cards are 'four of a kind'.
The number of ways of choosing the first 4 cards = 12
Number of ways of choosing the remaining card = 52 - 4 = 48
Hence the required number of 5-card hands = 12 x 48 = 576

7. Number of ways of choosing the consecutive numbers = 9
Number of ways of choosing the suit = 4
Hence the required number of 5-card hands = 9 x 4 = 36

8.(a) The required number of ways = 5! = 120

(b) Whether the arrangement is ended with 'T' or 'O', we both have 4! = 24 possibilities.
Hence the number of ways = 24 x 2 = 48

(c) 'O' and 'U' must be adjacent to each other.
That means, there must be either 'OU' or 'UO'.

First, we have to choose whether there is 'OU' or 'UO'. Obviously there are 2 ways to choose.

Next, we have to choose where the 'OU' or 'UO' to be put. There are 4 ways to choose.

At last, we have to choose the other 3 letters. There are 3! = 6 ways.

Hence there are 2 x 4 x 6 = 48 ways.

(d) There must be exactly one letter between 'U' and 'T'.
That means, there must be either 'U?T' or 'T?U'.

First, we have to choose whether there is 'U?T' or 'T?U'. Obviously there are 2 ways to choose.

Next, we have to choose where the 'U?T' or 'T?U' to be put. There are 3 ways to choose.

At last, we have to choose the other 3 letters. There are 3! = 6 ways.

Hence there are 2 x 3 x 6 = 36 ways.

9. The possible arrangements are:
(1) BGBGBGBGBG
(2) GBBGBGBGBG
(3) GBGBBGBGBG
(4) GBGBGBBGBG
(5) GBGBGBGBBG
(6) GBGBGBGBGB
Number of ways to choose the positions of boys and girls = 6

Number of ways to arrange the 5 boys among the 5 positions = 5! = 120
Similarly, number of ways to arrange the 5 girls among the 5 positions = 120

Hence the required number of ways = 6 x 120 x 120 = 86 400