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 26 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?
To determine the number of possible outcomes when your friend grabs 4 chocolates from the snack jar, we can use the concept of combinations.
In this scenario, your friend is selecting 4 chocolates out of a total of 26 (25 chocolates and 1 cashew). Since the order in which he selects the chocolates does not matter, we can use the combination formula.
The combination formula is given by:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of objects and r is the number of objects you want to select.
In this case, n = 26 (25 chocolates + 1 cashew) and r = 4 (the number of chocolates your friend will grab).
Plugging the values into the formula, we get:
C(26, 4) = 26! / (4! * (26 - 4)!)
= 26! / (4! * 22!)
Now, we can simplify this expression using the factorial notation:
26! = 26 * 25 * 24 * 23 * 22!
Substituting the values:
C(26, 4) = (26 * 25 * 24 * 23 * 22!) / (4! * 22!)
Here, the 22! terms in the numerator and denominator will cancel out.
C(26, 4) = (26 * 25 * 24 * 23) / (4!)
Evaluating the factorial:
4! = 4 * 3 * 2 * 1 = 24
Substituting the value:
C(26, 4) = (26 * 25 * 24 * 23) / 24
Finally, canceling out the 24 in the numerator and denominator:
C(26, 4) = 26 * 25 * 23 = 150,700
Therefore, there are 150,700 possible outcomes when your friend grabs 4 chocolates from the jar.