A tray contains 14 chocolate chip cookies and 16 oatmeal raisin cookies. How many ways can a child select 4 cookies that include at least 1 chocolate chip cookie?
To find the number of ways a child can select 4 cookies that include at least 1 chocolate chip cookie, we can consider two scenarios:
Scenario 1: Selecting exactly 1 chocolate chip cookie:
In this scenario, we choose 1 chocolate chip cookie from the 14 available cookies and 3 remaining cookies from the remaining 30 cookies (16 oatmeal raisin cookies and 14 chocolate chip cookies).
Number of ways = 14C1 * 30C3 = 14 * (30! / (3! * (30-3)!)) = 14 * (30! / (3! * 27!)) = 14 * (30 * 29 * 28) / (3 * 2 * 1) = 14 * 30 * 29 * 28 / 6 = 14 * 43520 = 609,280.
Scenario 2: Selecting more than 1 chocolate chip cookie:
In this scenario, we choose 2, 3, or 4 chocolate chip cookies from the 14 available cookies and select the remaining cookies from the rest.
Number of ways = (14C2 * 30C2) + (14C3 * 30C1) + (14C4 * 30C0)
= (14 * (14-1) / (2 * 1)) * (30 * (30-1) / (2 * 1)) + (14 * (14-1) * (14-2) / (3 * 2 * 1)) * (30 * (30-1) / (1)) + (14 * (14-1) * (14-2) * (14-3) / (4 * 3 * 2 * 1)) * (1)
= (91 * 435) + (364 * 29) + (273 * 30)
= 39,585 + 10,556 + 8,190
= 58,331.
Total number of ways = Scenario 1 + Scenario 2
= 609,280 + 58,331
= 667,611.
Therefore, there are 667,611 ways a child can select 4 cookies that include at least 1 chocolate chip cookie from the given tray.
To solve this problem, we'll use the concept of combinations, specifically the principle of inclusion-exclusion.
Step 1: Calculate the total number of ways to select any 4 cookies from the tray without any restrictions.
To do this, we will use the formula for combinations: nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items being chosen.
So, the total number of ways to select any 4 cookies from this tray (without any restrictions) is:
Total_combinations = (14 + 16)C4 = (30)C4 = 30! / (4! * (30-4)!) = 27,405
Step 2: Calculate the number of ways to select 4 cookies that have no chocolate chip cookies.
Since we want to find the number of ways with at least 1 chocolate chip cookie, we'll calculate the complement of this scenario.
The number of ways to select 4 cookies with no chocolate chip cookies is:
Non_CC_combinations = 16C4 = 16! / (4! * (16-4)!) = 4,368
Step 3: Calculate the number of ways to select 4 cookies with at least 1 chocolate chip cookie.
To find this, we'll subtract the number of ways to select 4 cookies with no chocolate chip cookies from the total number of ways to select any 4 cookies.
Ways_with_CC = Total_combinations - Non_CC_combinations
= 27,405 - 4,368
= 23,037
Therefore, there are 23,037 ways for the child to select 4 cookies from the tray with at least 1 chocolate chip cookie.
54
Chocolate cookie -- C
Oatmeal cookie ----O
If the chocolate cookies are the same and the oatmeal cookies are the same ...
there are only the following selections
COOO
CCOO
CCCO
CCCC
If the cookies are different:
1C,3O --- C(14,1)xC(16,3) = 7840
2C,2O --- C(14,2)xC(16,2) = 10920
3C,1O --- C(13,3)xC(16,1) = 4576
4C ------C(13,4)xC(16,0) = 715
for a total of 24051