Question-29

A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?

• 4
• 20
• 24
• 48

NoteAnswer

• 4
• 20
• 24
• 48

NoteSoluton

Let the number of scoops of ice-cream purchased of type \(\displaystyle i\) be \(\displaystyle x_{i}\). Then, we have:

\[ x_{1} +x_{2} +x_{3} +x_{4} =3 \]

where \(\displaystyle 0\leqslant x_{i} \leqslant 3\). The number of solutions to this equation is the same as the number of ways of purchasing ice-creams. The number of solutions is given by \(\displaystyle \binom{3+3}{3} =20\).