Question

Sheldon Pikyeeter has peculiar eating habits. For dinner, he always eats spaghetti 3 times per week, cheese pizza twice a week, Thai chicken one time per week, and tacos once a week. Note: for all questions below, consider each week as starting on Sunday and ending Saturday!

a) For a particular week, Sheldon is going to list all the possible different dinner plans. (Sunday through Saturday). How many weekly dinner plans are on Sheldon’s list?

b) What is the probability that a weekly dinner plan has spaghetti on Monday and pizza on Friday?

c) What is the probability that a weekly dinner plan has taco night before all of the spaghetti nights?

d) What is the probability that all of the spaghetti nights are on consecutive nights in a week?

e) What is the probability that none of the spaghetti dinners are on consecutive nights?

Answer 1

Answer a: Number of ways of eating spaghetti in a week= 3!

Number of ways of eating Cheese pizza in a week= 2!

Number of ways of eating Thai chicken in a week= 1!

Number of ways of eating tacos in a week= 1!

So total number of ways starting from sunday= 7 x 6 x 5 x4 x3 x 2x1

= 5040 plans

Answer b: probability that a weekly dinner plan has spaghetti on Monday and pizza on Friday+

we can write this as : Su Mo Tu Wed Thu Fri sat

sp piz

Dinner possibilities= 5 1 4 3 2 1 1

So we can say on sunday 5 kinds of dinner item possibilities and then on tuesday 4, wednesday 3, thusday 2 and saturday 1.

As on monday and friday 1 item each of spaghetti and pizza are fixed so,

Possible number of plans= 5x1x4x3x2x1x1 = 120

so probability = Favorable outcomes/ sample space = 120/5040 = .024

Answer C:

For getting Taco nights before spaghetti nights we can arrange weekly dinner plans as below:

cases Su Mo Tu We Th Fr Sat   

Taco on sunday Ta 6 5 4 3 2 1 = 6! ways= 720

Taco on Monday 3 Ta 5 4 3 2 1 = 3x1x 5x 4x3 x2 x1 = 360

Taco on Tuesday 3 2 Ta 4 3 2 1 = 3x2 x 1 x 4x 3x2x1 = 144

Taco on wednesday 3 2 1 Ta 3 2 1 = 3 x 2x 1 x1 x3 x2x 1= 36

Total possibilities= 1260

Probability= 1260/ 5040 = .25

Answer d : Probability for spaghetti nights on consecutive nights=

Cases Su Mo Tu We Th Fr Sa

1 sp sp sp

3 2 1 4 3 2 1 = 3x2x1x4x3x2x1 = 144

2 sp sp sp

4 3 2 1 3 2 1= 4x3x2x1x3x2x1= 144

3 sp sp sp

4 3 3 2 1 2 1 = 4x3x3x2x1x2x1 = 144

4 sp sp sp

4 3 2 3 2 1 1= 144

5 sp sp sp

4 3 2 1 3 2 1 = 144

So favorable outcomes = 144x5 = 720

Hence probability = 720/5040 = .1429

Answer E : Probability that none of spaghetti are on consecutive nights = 1- probability( they are on consecutive nights)

   = 1- .1429

   =0.8571