Permutations and Combinations: Arrangement of Identical Cans

Since you haven’t specified which problem you would like me to solve, I will solve Problem 10, as it is circled in your image and has handwritten notes.

Three identical cans of cola ( ), 2 identical cans of green tea ( ), and 2 identical cans of orange juice ( ) are arranged in a row. Calculate the number of arrangements if:
(i) the first and last cans in the row are the same type of drink.
(ii) the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.

Total number of cans = cans.

We consider three mutually exclusive cases based on which type of drink is at both ends.

Case 1: Both ends are Cola ( )
Remaining cans to arrange in the middle: (Total 5 cans).
Number of arrangements = .

Case 2: Both ends are Green Tea ( )
Remaining cans to arrange in the middle: (Total 5 cans).
Number of arrangements = .

Case 3: Both ends are Orange Juice ( )
Remaining cans to arrange in the middle: (Total 5 cans).
Number of arrangements = .

Total for (i) .

Step 1: Treat the 3 colas as a single block.
Let the block of colas be .
The items to arrange are now: (Total 5 items).

Step 2: Calculate arrangements where is present, ignoring the restriction on .
Total arrangements of .

Step 3: Subtract arrangements where the 2 green teas ( ) are next to each other.
Treat the two green teas as a single block .
The items to arrange are now: (Total 4 items).
Number of arrangements of .

Step 4: Final calculation for (ii).
Number of arrangements = (Total with ) (Arrangements where are together)
Number of arrangements .

Final Answers:
(i) 50
(ii) 18