Section 3.3

2) Five cards are dealt off of a standard 52-card deck and lined up in a row. How many such lineups are there in which all 5 cards are of the same suit?

(spade) + (hearts) + (diamonds) + (clubs)

1. specify a spade card for pos 1 (13)

2. specify an unseen spade card for pos 2 (12)

3. specify an unseen spade card for pos 3 (11)

4. specify an unseen spade card for pos 4 (10)

5. specify an unseen spade card for pos 5 (9)

   (13 x 12 x 11 x 10 x 9) x 4

4) Five cards are dealt off of a standard 52-card deck and lined up in a row. How many such lineups are there in which all 5 cards are of the same color (i.e., all black or all red)?

(red list) + (black list)

1. specify a red card for pos 1 (26)
2. specify an unseen red card for pos 2 (25)
3. specify an unseen red card for pos 3 (24)
4. specify an unseen red card for pos 4 (23)
5. specify an unseen red card for pos 5 (22)

(26 x 25 x 24 x 23 x 22) x 2

6) Consider lists made from the symbols A, B, C, D, E, with repetition allowed. (a) How many such length-5 lists have at least one letter repeated? (b) How many such length-6 lists have at least one letter repeated?

|A| = |B| + |C| ==> |A| - |C| = |B|

(all length 5 lists) = (length 5 lists w/ no repeats) + (length 5 lists with at least one repeat)

(all length 5 lists) - (length 5 lists w/ no repeats) = (length 5 lists with at least one repeat)

$5^5$ - 5 x 4 x 3 x 2 x 1

$5^6$ - 5 x 4 x 3 x 2 x 1 x 0

8) This problem concerns lists made from the letters A, B, C, D, E, F, G, H, I, J. (a) How many length-5 lists can be made from these letters if repetition is not allowed and the list must begin with a vowel? (b) How many length-5 lists can be made from these letters if repetition is not allowed and the list must begin and end with a vowel? (c) How many length-5 lists can be made from these letters if repetition is not allowed and the list must contain exactly one A?

1. Specify first pos as vowel (3)
2. Specify unseen letter for pos 2 (9)
3. Specify unseen letter for pos 3 (8)
4. Specify unseen letter for pos 4 (7)
5. Specify unseen letter for pos 5 (6)

3 x 9 x 8 x 7 x 6

1. Specify first pos as vowel (3)

2. Specify last pos as unseen vowel (2)

3. Specify unseen letter for pos 2 (8)

4. Specify unseen letter for pos 3 (7)

5. Specify unseen letter for pos 4 (6)

   3 x 2 x 8 x 7 x 6

For A in first pos:

1. Select A for pos 1 (1)

2. Specify unseen letter for pos 2 (9)

3. Specify unseen letter for pos 3 (8)

4. Specify unseen letter for pos 4 (7)

5. Specify unseen letter for pos 5 (6)

   Do same for pos 2, 3, 4, 5.

   (1 x 9 x 8 x 7 x 6) x 5

6. Choose pos for A (5)

7. Choose unseen letter for leftmost empty position (9)

8. Choose unseen letter for leftmost empty position (8)

9. Choose unseen letter for leftmost empty position (7)

10. Choose unseen letter for leftmost empty position (6)

10) Consider the lists of length six made with the symbols P, R, O, F, S, where repetition is allowed. (For example, the following is such a list: (P,R,O,O,F,S).) How many such lists can be made if the list must end in an S and the symbol O is used more than once?

(all ending in S) = (all ending in S with >= 2 Os) + (all ending in S with 1 Os) + (all ending in S no Os)

12) Six math books, four physics books and three chemistry books are arranged on a shelf. How many arrangements are possible if all books of the same subject are grouped together?

1. Select first topic (3)
2. Select second topic (2)
3. Select first math book (6)
4. Select second math book (5)
5. ...
6. Select first physics book (4)

...