A tray contains 11 chocolate chip cookies and 15 oatmeal raisin cookies. How many ways can a child select 4 cookies that include at least 1 chocolate chip cookie?

number without any restriction

= C(26,4) = 14950

number which are all oatmeal
= C(15,4) = 1365

so, the number with at least one chocolate chip
= 14950-1365 = 13585

Well, let's see. To find the number of ways a child can select 4 cookies with at least 1 chocolate chip cookie, we can break it down into different scenarios:

Scenario 1: The child selects 1 chocolate chip cookie and 3 oatmeal raisin cookies.
There are 11 ways to select 1 chocolate chip cookie and 15 choose 3 (or 15C3) ways to select 3 oatmeal raisin cookies.

Scenario 2: The child selects 2 chocolate chip cookies and 2 oatmeal raisin cookies.
There are 11 choose 2 (or 11C2) ways to select 2 chocolate chip cookies and 15 choose 2 (or 15C2) ways to select 2 oatmeal raisin cookies.

Scenario 3: The child selects 3 chocolate chip cookies and 1 oatmeal raisin cookie.
There are 11 choose 3 (or 11C3) ways to select 3 chocolate chip cookies and 15 ways to select 1 oatmeal raisin cookie.

Scenario 4: The child selects all 4 chocolate chip cookies.
There is only 1 way to select all 4 chocolate chip cookies.

Adding up all the possible scenarios, we get:

(11C1 * 15C3) + (11C2 * 15C2) + (11C3 * 15C1) + 1

And if you're brave enough to calculate all that, you'll find the answer!

To find the number of ways a child can select 4 cookies that include at least 1 chocolate chip cookie, we can use the concept of combinations.

Step 1: Calculate the total number of ways to select 4 cookies from the tray of 26 cookies.
We can use the combinations formula to find the total number of ways to select 4 cookies from a set of 26 cookies.
Total combinations = nCr = 26C4 = (26!)/[(4!)(26-4)!] = (26*25*24*23)/(4*3*2*1) = 14,950

Step 2: Calculate the number of ways to select 4 cookies without any chocolate chip cookies.
Since there are 15 oatmeal raisin cookies, the child can select all 4 cookies from the oatmeal raisin cookies, resulting in only 1 way to select 4 cookies without any chocolate chip cookies.

Step 3: Calculate the number of ways to select 4 cookies with at least 1 chocolate chip cookie.
To find this, subtract the number of ways to select 4 cookies without any chocolate chip cookies from the total number of combinations.
Number of ways with at least 1 chocolate chip cookie = Total combinations - Number of ways without any chocolate chip cookies
= 14,950 - 1
= 14,949

Therefore, there are 14,949 ways a child can select 4 cookies that include at least 1 chocolate chip cookie.

To calculate the number of ways a child can select 4 cookies that include at least 1 chocolate chip cookie, we can use the concept of combinations.

Step 1: Calculate the total number of ways to select any 4 cookies from the tray without restrictions.
This can be done using the combination formula: C(n, r) = n! / (r! * (n-r)!), where n is the total number of cookies and r is the number of cookies to be selected.

In this case, we need to select any 4 cookies from a tray that holds a total of 11 (chocolate chip) + 15 (oatmeal raisin) = 26 cookies. Using the combination formula, we get:
C(26, 4) = 26! / (4! * (26-4)!) = 26! / (4! * 22!)

Step 2: Calculate the number of ways to select 4 oatmeal raisin cookies.
Since we want to exclude the possibility of selecting only oatmeal raisin cookies, we need to calculate the number of ways to select 4 oatmeal raisin cookies from the tray, which is:
C(15, 4) = 15! / (4! * (15-4)!) = 15! / (4! * 11!)

Step 3: Calculate the number of ways to select 4 cookies without any chocolate chip cookies.
This can be found by subtracting the number of ways to select 4 oatmeal raisin cookies from the total number of ways to select any 4 cookies:
Total ways without chocolate chip = C(26, 4) - C(15, 4)

Step 4: Calculate the number of ways to select 4 cookies that include at least 1 chocolate chip cookie.
Since we want to include at least 1 chocolate chip cookie, we subtract the number of ways to select 4 cookies without any chocolate chip cookies from the total number of ways to select any 4 cookies:
Total ways with at least 1 chocolate chip = C(26, 4) - (C(26, 4) - C(15, 4))

Now, let's calculate the values:

C(26, 4) = 26! / (4! * 22!) = (26 * 25 * 24 * 23) / (4 * 3 * 2 * 1) = 13,455
C(15, 4) = 15! / (4! * 11!) = (15 * 14 * 13 * 12) / (4 * 3 * 2 * 1) = 1,365

Total ways without chocolate chip = C(26, 4) - C(15, 4) = 13,455 - 1,365 = 12,090
Total ways with at least 1 chocolate chip = C(26, 4) - (C(26, 4) - C(15, 4)) = 13,455 - 12,090 = 1,365

Therefore, there are 1,365 ways for the child to select 4 cookies that include at least 1 chocolate chip cookie from the given tray.