Question

Mark has 5 pebbles and 3 fish tanks. If each fish tank must have at least one pebble, how many different ways are there for Mark to put the pebbles in tanks if:

(a) The pebbles are identical, and the tanks are identical
(b) The pebbles are unique, and the tanks are identical
(c) The pebbles are unique, and each tank has a unique sticker
(d) The pebbles are identical, and each tank has a unique sticker Briefly justify your answer

Answer 1

a) 21 ways

(b) 243 ways

(c) 60 ways

(d) 60 ways

Step-by-step explanation:

a) The pebbles are identical, and the tanks are identical:

In this case, the only thing that matters is how many pebbles are in each tank, not the specific arrangement of pebbles. This is a classic "stars and bars" problem. Think of the pebbles as stars and the dividers between the tanks as bars. Mark has 5 pebbles and 3 tanks, so he needs 2 dividers to separate the tanks. The total number of ways to arrange the pebbles and dividers is then given by the binomial coefficient C(n + r - 1, r - 1), where n is the number of pebbles (5), r is the number of tanks (3). So, the answer is C(5 + 3 - 1, 3 - 1) = C(7, 2) = 21 ways.

(b) The pebbles are unique, and the tanks are identical:

Since the tanks are identical, the only thing that matters is which tank each pebble goes into. For each pebble, there are 3 choices (as there are 3 tanks). Since there are 5 pebbles, the total number of ways is 3^5 = 243 ways.

(c) The pebbles are unique, and each tank has a unique sticker:

In this case, both the pebbles and the tanks are unique, so the number of ways is simply the permutation of the pebbles taken 3 at a time (since there are 3 tanks). So, the answer is P(5, 3) = 5! / (5-3)! = 60 ways.

(d) The pebbles are identical, and each tank has a unique sticker:

This scenario is similar to (c), but the pebbles are identical. Therefore, the result is the same as in (c) since the identical nature of the pebbles does not affect the count when each tank has a unique sticker.

Answer 2

a) 21, b) 243, c) 6, d) 243

Step-by-step explanation:

(a) If the pebbles and tanks are identical, we can use the stars and bars method to solve this problem. We can think of each pebble as a star and the dividers between the tanks as bars. We need to distribute the 5 pebbles among the 3 tanks, so we need 2 dividers. The number of ways to arrange the stars and bars is given by the formula:

Number of ways = (5 + 2) choose 2 = 7 choose 2 = 21.

(b) If the pebbles are unique but the tanks are identical, we can use the concept of combinations. Each pebble can be placed in any one of the 3 tanks, so we have 3 choices for each pebble. The total number of ways to distribute the pebbles is:

Number of ways = 3 * 3 * 3 * 3 * 3 = 3^5 = 243.

(c) If the pebbles and tanks are unique, and each tank has a unique sticker, we can think of this as a permutation problem. The first pebble has 3 choices for tank, the second pebble has 2 choices, and so on. The total number of ways to distribute the pebbles is:

Number of ways = 3 * 2 * 1 = 6.

(d) If the pebbles are identical and each tank has a unique sticker, we can use the concept of combinations. Each pebble can be placed in any one of the 3 tanks, so we have 3 choices for each pebble. The total number of ways to distribute the pebbles is:

Number of ways = 3 * 3 * 3 * 3 * 3 = 3^5 = 243.