Counting Technique: rcombinations with repetition allowed

1
Counting Techniquer-combinations with
repetition allowed
2
Number of iterations of a nested loop(First
Situation)

• Consider the following nested loop
• for i1 to n
• for j1 to i-1
• for k1 to j-1
• Statements
• next k
• next j
• next i
• Question How many times the statements in the
  innermost loop will be executed?
• Solution Each iteration corresponds to
• a triple of integers (i, j, k) where i gt j gt
  k .
• The set of all this kind of triples corresponds
  to
• all 3-combinations of 1, , n .
• Thus, the total number of iterations is C(n,3)
  .

3
Number of iterations of a nested loop(Second
Situation)

• Consider the following nested loop
• for i1 to n
• for j1 to i
• for k1 to j
• Statements
• next k
• next j
• next i
• Question How many times the statements in the
  innermost loop will be executed?
• Solution Each iteration corresponds to
• a triple of integers (i, j, k) where i j
  k .
• Examples (5, 3, 2), (4, 4, 3), (2, 1, 1), (3,
  3, 3) .
• How to count the number of this kind of
  triples?

4
Number of iterations of a nested loop(Second
Situation)

• Each triple corresponds to a string of crosses
  and vertical bars.
• Numbers 1, 2, , n are considered as categories
• n-1 vertical bars separate the categories
• 3 crosses indicate
• how many items from each category are chosen.
• Examples when n5

5
Number of iterations of a nested loop(Second
Situation)

• Each triple corresponds to
• a string of n-1 vertical bars and 3 crosses.
• The length of any string is (n-1)3 n2 .
• The number of distinct positions
• for the 3 crosses in a string is C(n2, 3) .
• Thus, the number of distinct triples is C(n2,
  3) .
• Generalizing,
• if the number of nested loops is r
• then the number of iterations is C(n-1r, r) .

6
r-combinations with repetition allowed

• Definition
• r-combinations with repetition allowed
• from a set X of n elements
• is an unordered selection of r elements
  taken from X with repetition allowed.
• The number of r-combinations with repetition
  allowed from a set of n elements is C(n-1r, r) .

7
Which formula to use?

• We discussed four different ways
• of choosing r elements from n .
• The summary of formulas
• used in the four situations

8
Useful formulas for special cases

• C(n,n-r) C(n,r) .
• Proof
• C(n,n)1
• C(n,n-1) C(n,1) n
• C(n,n-2) C(n,2) n(n-1) / 2