Question 486535
there are 5 different types of flowers.  How many vases can be filled by using combinations of just 3 flowers without duplicating the 3 flowers previously used?


To do this, we need to use a combination of 3 flowers from 5 flowers, or 5C3, which calculates to: {{{(5!)/(3!2!)}}} ----- {{{(5*4*3!)/(3!2!)}}} ------ {{{(5*2cross(4)*cross(3!))/(cross(3!)cross(2!))}}} ----- {{{(5*2)/1}}} = 10 vases.


Hypothetically, suppose these 5 different flowers also had different colors. We could name them as follows: R (red), W (white), P (pink), Y (yellow), and B (blue)


These are the different arrangements:


Vase  1: RWP
Vase  2: RWY
Vase  3: RWB
Vase  4: RPY
Vase  5: RPB
Vase  6: RYB
Vase  7: WPY
Vase  8: WPB
Vase  9: WYB
Vase 10: PYB


As seen, each vase will have a different 3-color combination. As such, there are {{{highlight_green(10)}}} different combinations.