Generalized Permutations

1
Generalized Permutations Combinations
Selected Exercises
2
10 (a)

• A croissant shop has plain croissants, cherry
  croissants, chocolate croissants, almond
  croissants, apple croissants, broccoli
  croissants.
• How many ways are there to choose 12 croissants?
• Abstract version of the problem
• There is an infinite supply of 6 kinds of
  objects.
• How many ways are there to choose 12 of them?

3
10 (a) Solution

• How many binary strings of length 12 6 1 are
  there with exactly 12 0s? ( of length-17
  binary strings with 5 1s.)
• C( 12 6 1, 6 1 ) C( 12 6 1, 12 ).
• How many ways are there to order 12 items from a
  menu of 6 kinds of items?
• 1-to1 correspondence between binary strings and
  orders.

Item 1 Item 2 Item 3 Item 4 Item 5 Item 6

4
10 (b)

• A croissant shop has plain croissants, cherry
  croissants, chocolate croissants, almond
  croissants, apple croissants, broccoli
  croissants.
• How many ways are there to choose 36 croissants?
• Abstract version of the problem
• There is an infinite supply of 6 kinds of
  objects.
• How many ways are there to choose 36 of them?

5
10 (b) Solution

• How many binary strings of length 36 6 1 are
  there with exactly 36 0s? ( of length-41
  binary strings with 5 1s.)
• C( 36 6 1, 6 1 ) C( 36 6 1, 36 ).
• How many ways are there to order 36 items from a
  menu of 6 kinds of items?
• 1-to1 correspondence between binary strings and
  orders.

6
10 (c)

• A croissant shop has plain croissants, cherry
  croissants, chocolate croissants, almond
  croissants, apple croissants, broccoli
  croissants.
• How many ways are there to choose 24 croissants
  with 2 of each kind?
• Abstract version of the problem
• There is an infinite supply of 6 kinds of
  objects.
• How many ways are there to choose 24 of them with
  2 of each kind?

7
10 (c) Solution

• How many ways are there to order 24 2 . 6 12
  items from a menu of 6 kinds of items?
• There is a 1-to-1 correspondence between these
  orders and the original type of orders
• For each order of 12 items
• For each kind of item, increment the order of
  that kind by 2.
• The resulting order has 24 items, 2 of each
  kind of item.
• Answer C( 12 6 - 1, 6 1 ).

8
10 (d)

• A croissant shop has plain croissants, cherry
  croissants, chocolate croissants, almond
  croissants, apple croissants, broccoli
  croissants.
• How many ways are there to choose 24 croissants
  with ? 2 broccoli?
• Abstract version of the problem
• There is an infinite supply of 6 kinds of
  objects.
• How many ways are there to choose 24 of them with
  ? 2 of kind 1?

9
10 (d) Solution

• We can use the sum rule to decompose this problem
  into 3 sub-problems, based on an exact of
  broccoli croissants
• Count the solutions with exactly 0 broccoli
  croissant
• C( 24 5 - 1, 5 1 ).
• 2. Count the solutions with exactly 1 broccoli
  croissant
• Pick the broccoli croissant 1
• Pick the 23 remaining croissants from the
  remaining 5 kinds of croissants C( 23 5 - 1, 5
  1 ).
• 3. Count the solutions with exactly 2 broccoli
  croissants
• Pick the 2 broccoli croissants 2
• Pick the 22 remaining croissants from the
  remaining 5 kinds of croissants C( 22 5 - 1, 5
  1 ).

10
10 (d) Better Solution

• Count all orders of 24 croissants
• C( 24 6 1, 6 1 )
• Subtract the bad orders
• Order 21 other croissants from all 6 varieties
• C( 21 6 1, 6 1 )
• Answer C( 24 6 1, 6 1 ) C( 21 6 1, 6
  1 )

11
10 (e)

• A croissant shop has plain croissants, cherry
  croissants, chocolate croissants, almond
  croissants, apple croissants, broccoli
  croissants.
• How many ways are there to choose 24 with 5
  chocolate 3 almond?
• Abstract version of the problem
• There is an infinite supply of 6 kinds of
  objects.
• How many ways are there to choose 24 of them with
  5 of kind 1 3 of kind 2?

12
10 (e)

• There is a 1-to-1 correspondence between
• orders of 24 5 3 objects of 6 kinds
• Orders of 24 objects of 6 kinds with 5 of kind
  1 3 of kind 2.
• Correspondence Add
• 5 objects of kind 1
• 3 objects of kind 2
• to each order of type 1 to get an order of type
  2.
• We count the of orders of type 1
• Answer C( 24 5 3 6 1, 6 1 )

13
10 (f)

• A croissant shop has plain croissants, cherry
  croissants, chocolate croissants, almond
  croissants, apple croissants, broccoli
  croissants.
• How many ways are there to choose 24 with
• 1 plain 2 cherry 3 chocolate 1
  almond 2 apple
• ? 3 broccoli?
• Abstract version of the problem
• There is an infinite supply of 6 kinds of
  objects.
• How many ways are there to choose 24 of them
  with
• 1 of kind 1 2 of kind 2 3 of kind 3
  1 of kind 4 2 of kind 5
• ? 3 of kind 6

14
10 (f)

• Put 1 plain 2 cherry 3 chocolate 1 almond
  2 apple to the side (9 objects to the side)
• There are 24 9 15 left to distribute w/o
  restriction on broccoli.
• C( 15 6 1, 6 1 )
• 3. Subtract the bad orders ( 4 broccoli)
• C( 11 6 1, 6 1)
• Answer C( 15 6 1, 6 1 ) C( 11 6 1, 6
  1)

15
20

• How many integer solutions are there to the
  inequality
• x1 x2 x3 ? 11, for x1 , x2 , x3 0?
• Hint Introduce an auxiliary variable x4 such
  that
• x1 x2 x3 x4 11.

16
20 Solution

• There is a 1-to-1 correspondence between
• solutions to the equality
• solutions to the inequality question
• The values of x1 , x2 , x3 from the equality
  solve the inequality.
• Equivalent to counting solutions to the equality
• How many ways are there to order 11 items from a
  menu of 4 kinds of items?
• Answer C( 11 4 1, 4 1 ).

17
30

• How many different strings can be made from the
  letters in MISSISSIPPI, using all 11 letters?

18
30 Solution

• Using the product rule
• Pick the position in the 11-letter string where
  the letter M goes C( 11, 1 )
• Pick the position in the 10 remaining positions
  where the 4 I letters go C( 10, 4 )
• Pick the position in the 6 remaining positions
  where the 4 S letters go C( 6, 4 )
• Pick the position in the 2 remaining positions
  where the 2 P letters go C( 2, 2 )
• Answer 11! / 1!4!4!2!.

19
30 Generally

• If you have n objects such that
• n1 objects of them are of type t1
• n2 objects of them are of type t2
• . . .
• nk objects of them are of type tk
• The of arrangements of these objects is
• C(n, n1 ) C( n - n1 , n2 ) C( n n1 n2 , n3 )
  C( n n1 nk-1, nk)
• n! / (n1! n2! nk!)
• (This equality is simple to verify algebraically.)

20
40

• How many ways are there to travel in xyzw space
  from the origin (0, 0, 0, 0) to (4, 3, 5, 4) by
  taking steps
• 1 unit in the positive x direction
• 1 unit in the positive y direction,
• 1 unit in the positive z direction
• 1 unit in the positive w direction?

21
40 Solution

• Any path from (0, 0, 0, 0) to (4, 3, 5, 4) is a
  sequence with
• 4 x steps
• 3 y steps
• 5 z steps
• 4 w steps.
• Equivalent problem How many 16 letter sequences
  of x, y, z, and w are there with exactly
• 4 x,
• 3 y
• 5 z
• 4 w ?
• Answer 16! / 4! 3! 5! 4!.

22
50

• How many ways are there to distribute 5
  distinguishable objects in 3 indistinguishable
  boxes?

23
50 Solution

• Use the sum rule to decompose the set of
  solutions into disjoint subsets
• solutions with 5 objects in 1 box 1.
• solutions with 4 objects in 1 box, 1 object in
  another C(5, 1).
• solutions with 3 objects in 1 box, 2 objects in
  a 2nd C(5, 3).
• solutions with 3 objects in 1 box, 1 object in
  a 2nd box, 1 object in a 3rd box C(5, 3).
• solutions with 2 objects in 1 box, 2 objects in
  a 2nd box, 1 object in a 3rd box C(5, 1) C(4,
  2).
• Answer 1 C(5, 1) C(5, 3) C(5, 3) C(5, 1)
  C(4, 2).

24
60

• Suppose a basketball league has 32 teams, split
  into 2 conferences of 16 teams each. Each
  conference is split into 3 divisions. Suppose
  that the North Central Division has 5 teams. Each
  of the teams in this division plays
• 4 games against each of the other teams in this
  division
• 3 games against each of the 11 remaining teams in
  the division, and
• 2 games against each of the 16 teams in the other
  conference.
• In how many different orders can the games of 1
  of the teams in the North Central Division be
  scheduled?

25
60 Solution

• Let the 4 other teams in the North Central
  Division be named x1, x2, x3, x4.
• Let the 11 other teams in the division be named
  y1, y2, , y11.
• Let the 16 teams in the other division be named
  z1, z2, , z16.
• The total of games that a team plays is
• 4 . 4 11 . 3 16 . 2 81
• The number of 81-letter sequences with4 each of
  x1, x2, x3, x4
• 3 each of named y1, y2, , y11
• 2 each of z1, z2, , z16 is
• 81! / (4!)4(3!)11(2!)16

26
Characters

• ? ? ? .
• ? ? ? ? ? ? ?
• ? ? ?
• ? ?? T
• ? ? ? ? ? S
• ? ? ? ? ? ? ? ?