Question

Toronto Men's Swimming Club is organizing a competition. How many ways are there to finish this race for 5 swimmer's if ≥ 2 swimmer's are allowed to be at the same position?

Answer 1

Let comma represent the number of swimmers passing the finish line at the same time.

• All pass together: 1 way
• 4, 1 or 1, 4: 5c4*2 = 10 ways
• 3, 2 or 2, 3
  = 5c3*2 = 20 ways
• 3, 1, 1 and its permutations:
  = 5c3*3! = 60 ways
• 2, 2, 1 and its permutations:
  = (5c2)*(3c2)*3!/2 = 90 ways
• 2, 1, 1, 1 and its permutations:
  = (5c2)*4! = 240 ways
• 1, 1, 1, 1, 1 that is 5! = 120 ways

Therefore total number of ways in which the swimmers can complete the race is computed here as:
= 1 + 10 + 20 + 60 + 90 + 240 + 120 = 541 ways

Therefore there are 541 ways to complete the race here.