Discussion
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?
Answer: B
There are 20 ways.
Since the order does not matter and a flavour may be repeated, a purchase is a selection of 3 scoops from 4 flavours with repetition allowed. The number of such selections is given by the stars-and-bars formula:
\[ \begin{aligned} \binom{n + r - 1}{r} &= \binom{4 + 3 - 1}{3} \\ &= \binom{6}{3} = 20 \end{aligned} \]
As a check: all three of one flavour gives 4 ways, two of one and one of another gives \(4 \times 3 = 12\), and three different flavours gives 4, a total of 20.
4 counts only the choices of three different flavours, 24 counts ordered choices of three different flavours, and 48 counts neither case correctly.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development