At Sally's Salads, there is an all you can eat salad bar. You may choose one of 3 different lettuces, any number of 11 vegetable toppings, and at most one of 6 dressings. Suppose a salad is made at random following those rules. If the salad includes a lettuce choice, but not necessarily anything else, what is the probability that you get a salad with 4 vegie toppings and Ranch dressing?
number of ways to choose 1 of 3 salads
= C(3,1) = 3
number of ways to choose any number of 11 toppings, including choosing no topping
= 2^11 = 2048
number of possible salads = 3(2048) = 6144
number of salads with Ranch and 4 dopping
= 1x C(11,4) = 330
prop of event = 330/6144
= 55/1024
To find the probability of getting a salad with 4 veggie toppings and Ranch dressing, we need to first determine the total number of possible salads and then calculate the number of salads that meet the given criteria.
Step 1: Calculate the Total Number of Possible Salads
There are 3 choices of lettuce: Lettuce 1, Lettuce 2, and Lettuce 3.
There are 11 vegetable toppings, and we can choose any number (from 0 to 11) of these toppings.
There are 6 dressings, and we can choose at most one of them.
To calculate the total number of possible salads, we need to consider all possible combinations of lettuce, veggie toppings, and dressing. We can do this by multiplying the number of choices for each category:
Total number of possible salads = (Number of lettuce choices) × (Number of veggie topping choices) × (Number of dressing choices)
Total number of possible salads = 3 × (2^11) × (7 choose 0 + 7 choose 1 + 7 choose 2 ... + 7 choose 11) × 6
Here, (2^11) represents the number of choices for the veggie toppings (either include or exclude each of the 11 toppings), and (7 choose 0 + 7 choose 1 + 7 choose 2 ... + 7 choose 11) represents all possible combinations of toppings (from 0 to 11).
Step 2: Calculate the Number of Salads with 4 Vegie Toppings and Ranch Dressing
For the salad to have 4 veggie toppings and Ranch dressing, we need to choose 4 out of the 11 veggie toppings and also choose Ranch dressing.
Number of salads with 4 veggie toppings and Ranch dressing = (Number of veggie topping combinations) × (Number of dressing choices)
Number of salads with 4 veggie toppings and Ranch dressing = (11 choose 4) × 1
Step 3: Calculate the Probability
Finally, we can calculate the probability by dividing the number of salads with 4 veggie toppings and Ranch dressing by the total number of possible salads:
Probability = (Number of salads with 4 veggie toppings and Ranch dressing) / (Total number of possible salads)
Probability = [(11 choose 4) × 1] / [3 × (2^11) × (7 choose 0 + 7 choose 1 + 7 choose 2 ... + 7 choose 11) × 6]
By evaluating the equation, you will get the probability of getting a salad with 4 veggie toppings and Ranch dressing at Sally's Salads.