Combinations with repetitions allowed

1
Combinations with repetitions allowed

• Red, green and blue cubes are given. There are at
  least five cubes of each of the above colors. In
  how many ways can a selection of five cubes be
  made, such that selecting zero or more cubes of
  the same color is allowed?
• This is the problem of combinations with
  repetitions allowed.

2

• There are n distinct types of objects. In how
  many ways can r objects be selected, such that
  each object is of one of the above n types, and
  zero or more objects of each of the above types
  may be selected?
• Solution Write down a line of n-1 bars.
• (For example if n3, the line will be ).
• Those bars create n spaces from left to
  right the space to the left of the leftmost bar,
  followed by the space between the first and the
  second bar, etc., and finally the space to the
  right of the
• rightmost bar.

3

• Then write zero or more stars in each space,
• such that the total number of stars is r,
• Thus creating a row of n-1 bars and r stars.
• Each combination as described above can be
• uniquely represented by such a row
• Number the n spaces 1,, n from left to right.
• Then the number of stars in the jth space
  represents the number of objects of type j
  selected.
• Example n3, r5,
• ? ? ? ? ? 1 object of type 1, 1 of type
  2, 3 of type 3
• ? ? ? ? ? zero objects of type 1, 1 object
  of type 2, 4 objects of type 3.

4

• So the number of combinations described above is
  equal to the number of rows of n-1 bars and r
  stars. Such a row has nr-1 characters, and is
  specified by selecting the locations of the r
  stars out of the nr-1 locations. Therefore, the
  number of such rows is C(nr-1,r), and this is
  also the number of combinations with repetitions
  allowed.
• In the above problem about red, green, and
  blue cubes, n3 and r5. The solution is
  C(nr-1,r)C(7,5)21.

5

• The problem of combinations with repetitions
  allowed with given n and r is equivalent to the
  following problem
• How many solutions are there to the equation
• x1 xn r
• such that each xi is a non negative integer?
• xi represents the number of objects of type
  i
• selected.

6

• Example How many solutions does the equation
• x1x2x3 5 have such that x1, x2, x3 are
  non negative integers.
• Solution Here n3 and r5, so the answer is
• C(nr-1,r)C(7,5)21.
• Example How many solutions does the equation
  x1x2x3 8 have such that x1, x2, x3 are
  integers, x1?1, x2 ? 0, x3 ?2 ?
• Solution Use the above correspondence between
  solutions of the equation and selecting r objects
  given n types of objects. Since 3 objects (one of
  type 1 and two of type 3 were already selected, 5
  more have to be selected. So r5, and the answer
• is C(nr-1,r)C(7,5)21.