My couch potato friend enjoys sitting in front of the TV and grabbing handfuls of 4 chocolates at random from his snack jar. Unbeknownst to him, I have replaced one of the 18 chocolates in his jar with a cashew. (He hates cashews with a passion.)
How many possible outcomes are there the first time he grabs 4 chocolates?
1
How many of these include the cashew?
To determine the number of possible outcomes the first time your friend grabs 4 chocolates, we need to use combinations. The formula for combinations is:
nCk = n! / (k!(n-k)!)
In this case, n represents the total number of chocolates in the jar (18) and k represents the number of chocolates your friend grabs (4).
So, the number of possible outcomes is:
18C4 = 18! / (4!(18-4)!)
= 18! / (4!14!)
Calculating this, we get:
18! = 18 x 17 x 16 x 15 x 14!
Thus,
18C4 = (18 x 17 x 16 x 15 x 14!) / (4!14!)
= (18 x 17 x 16 x 15) / (4 x 3 x 2 x 1)
= 3060
Therefore, there are 3060 possible outcomes the first time your friend grabs 4 chocolates.
To determine how many of these outcomes include the cashew, we need to consider that the cashew can be in one of the 4 chocolates your friend grabs. Since there is only 1 cashew in the jar, there are 4 possible positions for the cashew within the four chocolates.
So, the number of outcomes that include the cashew is:
4
Therefore, out of the 3060 possible outcomes, 4 of them include the cashew.