Question

You volunteer to help drive children at a charity event to the zoo, but you can fit only 6 of the 17 children present in your van How many different groups of 6 childrencan you drive?

Answer 1

Since we need to choose groups of 6 from 17 options, we call this "17 choose 6". This is the same as the nCr formula for "n choose r", so we need to use:

$\begin{gathered} C(n,r)=\frac{n!}{r!(n-r)!} \\ C(17,6)=\frac{17!}{6!(17-6)!}=\frac{17!}{6!\cdot11!} \end{gathered}$

Since factorial is the multiplication of the factors from the given one to 1, 17! divided by 11! cancels out all the factors from 11 to 1 of the 17!, so we are left with:

$C(17,6)=\frac{17!}{6!(17-6)!}=\frac{17!}{6!\cdot11!}=\frac{17\cdot16\cdot15\cdot14\cdot13\cdot12}{6\cdot5\cdot4\cdot3\cdot2\cdot1}$

Now, we can cancel some factors out before evaluating everthing:

$C(17,6)=\frac{17!}{6!(17-6)!}=\frac{17!}{6!\cdot11!}=\frac{17\cdot16\cdot15\cdot14\cdot13\cdot12}{6\cdot5\cdot4\cdot3\cdot2\cdot1}=\frac{17\cdot4\cdot1\cdot14\cdot13\cdot1}{1\cdot1\cdot1\cdot1\cdot1\cdot1}=17\cdot4\cdot14\cdot13=12376$

So, the number of combinations is 12376.