Question

Amy and Adam are making boxes of truffles to give out as wedding favours. They have an unlimited supply of 5 different types of truffles. If each box holds 2 truffles of different types, how many different boxes can they make?
(a) 12
(b) 10
(c) 15
(d) 20

Answer 1

Hint: In this question, we need to find the number of combinations that can be possible in which out of 5 we need to select 2 truffles and place them in the box which can be done using the combinations formula given by \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\] which on further simplification gives the result.

Complete step-by-step answer:
COMBINATION:
Each of the different groups or selections which can be made by some or all of a number of given things without reference to the order of the things in each group is called a combination
Mathematically the number of combinations of n different things taken r at a time is given by
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\]
Now, in the question given that there are 5 different truffles
Now, we need to place 2 different truffles in a box which can be done using the above combination formula
Now, on comparing the given values in the question with the above formula we have
\[n=5,r=2\]
Now, from the above combinations formula to place 2 truffles in a box out of 5 different truffles we have
\[\Rightarrow {}^{n}{{C}_{r}}\]
Now, on substituting the respective values we get,
\[\Rightarrow {}^{5}{{C}_{2}}\]
Now, this can be further written in the simplified form as
\[\Rightarrow \dfrac{5!}{2!\left( 5-2 \right)!}\]
Now, this can be further written as
\[\Rightarrow \dfrac{5!}{2!3!}\]
Now, on further writing them in expanded form we get,
\[\Rightarrow \dfrac{5\times 4\times 3\times 2\times 1}{2\times 1\times 3\times 2\times 1}\]
Now, on cancelling out the common terms we get,
\[\Rightarrow 5\times 2\]
Now, on further simplification we get,
\[\Rightarrow 10\]
Hence, the correct option is (b).

Note:Instead of using the combinations formula we can also solve it by using the permutations formula by considering the arrangement of 2 like things in 5 different things using the formula \[\dfrac{\left( m+n \right)!}{m!n!}\] which on simplification gives the same result.
It is important to note that as the number of boxes are not defined exactly we just need to select 2 truffles out of 5 truffles which gives the result and then substitute in the combinations formula.