Panther Prowl is an annual CEAS event. The college organizes a day of fun activi

Question

Panther Prowl is an annual CEAS event. The college organizes a day of fun activities and free food for all CEAS students. Free food typically consists of sandwiches from Jimmy Johns. Over the years, the college has narrowed down the choices to the Pepe (Ham), Turkey, Tom, Vito (Italian), Totally Tuna and Vegetarian. They plan to order 1000 sandwiches. Since we just learned how to count, we can't help it but turn this into a counting problem. How many different orders are possible if a. there should be at least 100 sandwiches for each choice? b. there should be 600 meat sandwiches (Pepe; Turkey Tom, or Vito) and 400 non-meat sandwiches (Totally Tuna or Vegetarian)? c. there should be at most 250 Vegetarian sandwiches d. there should be at least 200 Totally Tuna or at least 200 Vegetarian sandwiches?

Explanation / Answer

Let say Total number of sandwich of type

Pepe = x1, Turkey Tom = x2 , Vito = x3 and Totally Tuna = x4 and Vegeterian = x5

so The counting problem conditionis

X1 + X2 + X3 + X4 + X5 = 1000

(a) Here X1, X2,X3,X4,X5 >= 100

so Here Xi = Yi + 100 and so on

so Y1 + Y2 + Y3 + Y4 + Y5 = 500 where Yi >=0

so total combinations = (500 + 5 -1) C (5-1) = 504C4

(b) There should be 600 meat sandwitches and 400 non- meat

X1 + X2 + X3 = 600 ; X4 + X5 = 400

so Here total possiblities in first quation = (600 + 3 -1)C(3-1) = 602C2

In second equation = 401C1

so tota possiblities = 602C2 401C1

(c) There should be atleast 250 vegeterian sandwitches

so X5 >= 250

so X1 + X2 + X3 + X4 + Y5 = 750 [ where Y5 = X5 + 250]

Here Xi where i = 1,2,3,4 and Y5 >=0

so Total combinations = (750 + 5-1)C (5-1) = 754C4

(d) Here there are atleast 200 totally tuna and 200 atleast Begeterian sandwitches

so X1 + X2 + X3 + Y4 + Y5 = 600

[ where Y4 = X4 + 200 and Y5 = X5 + 200]

so Total possibilities or order = 604C4

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