Question

a food company is testing 5 chocolate chip cookies, 4 crackers, and 7 reduced - fat cookies. if it plans to market 1 of the chocolate chip cookies, 3 of the crackers, and 2 of the reduced - fat cookies, how many different combinations are possible?
there are \\( \square \\) possible combinations.
(type a whole number.)

Explanation:

Step1: Use combination formula

The combination formula is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n\) is the total number of items and \(r\) is the number of items to be chosen.
For chocolate - chip cookies: \(n_1 = 5\), \(r_1=1\), so \(C(5,1)=\frac{5!}{1!(5 - 1)!}=\frac{5!}{1!4!}=5\)

Step2: For crackers

\(n_2 = 4\), \(r_2 = 3\), so \(C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=4\)

Step3: For reduced - fat cookies

\(n_3=7\), \(r_3 = 2\), so \(C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7\times6\times5!}{2\times1\times5!}=21\)

Step4: Use the multiplication principle

The total number of combinations is the product of the number of combinations of each type. So the total number of combinations \(N = C(5,1)\times C(4,3)\times C(7,2)\)
Substitute the values: \(N=5\times4\times21\)
\(N = 420\)

Answer:

\(420\)