Question

Bar Management

Molson currently sells 41 different brands of beer in Canada. Labatt currently sells 17 different brands.

1. The manager at Mike's Place needs to choose 5 Molson brands and 5 Labatt brands to sell. How many options do they have?
2. The 10 beers (5 Labatt, 5 Molson) selected in the previous question must be placed in a line on a display shelf so that no two Molson products are adjacent and no two Labatt products are adjacent. How many ways are there to do this?
3. Continuing from the previous question, suppose two of the brands selected by the manager were (Labatt) 50 and (Molson) Export. The display shelf can not have a bottle of 50 adjacent to a bottle of Export. How many ways are there to do this while still avoiding two adjacent Molson products and two adjacent Labatt products?
4. How many of the arrangements from the previous question have the bottle of 50 and the bottle of Export among the first (leftmost) 5 bottles?

Answer 1

• Given that there are 41 Molson Brands and 17 Labatt Brands and we need to choose 5 of each we have different combinations of 10 beers to choose i.e. we choose 5 out of 41 Molson beers and for each of these choices we choose 5 Labatt brands using Combinations nCr
• We need to arrange these beers on a shelf in such a way such that No two Molson beers are adjacent and no two Labatt bottles are adjacent. Let us visualise this

  M

  L

  M

  L

  M

  L

  M

  L

  M

  L

  Now the Molston Brands can be rearranged in 5! ways and the Labatt brands can be arranged in also 5! ways so Total number of ways of arranging in above set up is 5!x5! . Another way to visualise this is

  L

  M

  L

  M

  L

  M

  L

  M

  L

  M

  Here again each type of beer can be rearranged in 5! ways and the total ways of arranging this is 5!x5! ways . So our final answer is TOTAL NUMBER OF ARRANGEMENTS = 2x5!x5! = 28800.
• Now we have specific Labatt and Molson Brands that cannot be adjacent. Refer to the arrangement figures in the last question. Case 1) Let us calculate the number of arrangements which will have the two specific brands ajacent to each other, let the first two blocks denote them . Fixing these two the rest of the bottles can be arranged in 4!x4! ways further the two adjacent bottles can have 5 positions so the total arrangements if 5x4!x4! = 2880. Case 2) Along similar lines here too there will be 2880 arrangements with Labatt 50 and Molson Export beer next to each other . So finally the number of arrangements where these two brands are not adjacent will be 28800-2880-2880 = 23040.
• We want the bottle of 50 and the bottle of export in the first 5 bottles under the arrangement of the previous question so they cannot be adjacent. So that will be possible in only four ways taking into account the previous two arrangements drawn

  M

  L(50)

  M

  L

  M(Exp)

  L

  M

  L

  M

  L

  and

  M(Exp)

  L

  M

  L(50)

  M

  L

  M

  L

  M

  L

  and

• L(50)

  M

  L

  M(Exp)

  L

  M

  L

  M

  L

  M

  and

  L

  M

  L(50)

  M

  L

  M(Exp)

  L

  M

  L

  M

  for each of these four arrangements the remaining three bottles of Labatt and Molston can be picked in 3! ways each , so the Arrangements where the 50 and export appear in the first 5 bottles but are not adjacent to each other is in 4x3!x3! = 144