Question

A customer at Marty’s Fruit Stand picks a sample of 3 oranges at random from a crate containing 60 oranges, of which 4 are rotten. In how many ways can you chose at least two rotten oranges?

Answer 1

Answer:

340 ways

Step-by-step explanation:

Given:

Total number of oranges = 60

Number of rotten oranges = 4

Number of oranges picked = 3

Now, number of good oranges = Total number - Rotten oranges

                                                    = 60 - 4 = 56

Now, we need to pick at least two rotten oranges.

So, the possible outcomes can be as follows:

1. 2 rotten oranges + 1 good orange = 3 oranges
2. 3 rotten oranges + 0 good orange = 3 oranges

Now, number of ways of picking 'r' distinct objects from a total of 'n' objects is given  as:

$^nCr=\frac{n!}{r!(n-r)!}$

Now, picking 2 rotten oranges from a total of 4 rotten oranges is:

$^4C_2=\frac{4!}{2!2!}=\frac{4\times 3\times 2}{4}=6$

Similarly, picking 3 rotten oranges from a total of 4 rotten oranges is:

$^4C_3 =\frac{4!}{3!\times1!}=\frac{4\times 3!}{3!}=4$

Now, picking 1 good orange from a total of 56 good oranges is:

$^{56}C_1=56$

Picking 0 good oranges means picking no good oranges.

Therefore, the total number of ways of picking at least 2 rotten oranges is the sum of the above two possibilities and is given as:

At least 2 rotten out of 3 picked = (2 rotten and 1 good) or 3 rotten

                                                       = 6 × 56 + 4

                                                       = 336 + 4 = 340 ways

Therefore, there are 340 ways of picking at least 2 rotten oranges when 3 oranges are picked from a total of 60 oranges.