A Finite Student Has A Set Of Differently Shaped Plastic Obj

A Finite Student Has A Set Of Differently Shaped Plastic Objects Ther

A finite student possesses a collection of distinctly shaped plastic objects, including 3 pyramids, 4 cubes, and 7 spheres. The following problems explore various arrangements and selections involving these objects, considering different constraints related to their shape, color, and order of selection.

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a) In how many ways can the objects be arranged in a row if each is a different color?

Since each object has a unique shape and is of a different color, all 14 objects (3 pyramids, 4 cubes, and 7 spheres) are distinct. The total number of arrangements (permutations) of these objects in a row, with each position occupied by a different object, is given by the factorial of the total number of objects:

\[ 14! = 14 \times 13 \times 12 \times \ldots \times 2 \times 1 \]

Calculating this yields:

\[ 14! = 87,178,291,200 \]\

b) How many arrangements are possible if objects of the same shape must be grouped together and each object is a different color?

In this scenario, objects are grouped by shape, and within each shape, objects are distinguishable by color. The objects form three groups: pyramids, cubes, and spheres.

• Number of ways to permute the shape groups: 3! = 6
• Within each group, objects can be arranged in factorial of the number of objects:
  • Pyramids: 3 objects, so 3! = 6
  • Cubes: 4 objects, so 4! = 24
  • Spheres: 7 objects, so 7! = 5040

Thus, the total number of arrangements is:

\[ 3! \times 3! \times 4! \times 7! = 6 \times 6 \times 24 \times 5040 = 4,366,080 \]

c) In how many distinguishable ways can the objects be arranged in a row if objects of the same shape are also the same color but need not be grouped together?

Here, objects of the same shape are considered identical and of the same color, reducing distinguishability within each shape. Since objects are of different colors, but the problem states objects of the same shape are the same color, we interpret this as treating all objects within each shape as identical. However, because each shape has multiple objects, the situation simplifies to selecting arrangements considering identical objects within each shape group.

Given that shapes are considered identical within shape categories and objects of the same shape are the same color, the total arrangements depend on permuting the objects considering identical objects within each shape group:

• Pyramids: 3 identical objects
• Cubes: 4 identical objects
• Spheres: 7 identical objects

Total objects: 14

Number of arrangements—since objects of the same shape are indistinct within their group—is computed as the multinomial coefficient:

\[ \frac{14!}{3! \times 4! \times 7!} \]

Calculating:

\[ \frac{14!}{3! \times 4! \times 7!} = \frac{87,178,291,200}{6 \times 24 \times 5040} \]

= \(\frac{87,178,291,200}{725,760} \approx 120,000\)

Therefore, there are approximately 120,000 distinguishable arrangements in this scenario.

d) In how many ways can you select 3 objects, one of each shape, if the order in which the objects are selected does not matter and each object is a different color?

Since each selected object is of a different shape, the selection involves choosing:

• 1 pyramid out of 3
• 1 cube out of 4
• 1 sphere out of 7

The total number of selections is the product of these individual choices:

\[ 3 \times 4 \times 7 = 84 \]

e) In how many ways can you select 3 objects, one of each shape, if the order in which the objects are selected matters and each object is a different color?

Here, the order of selection is significant. The number of ways to select ordered triples, with each object being distinct in shape and color, involves permutations:

• Number of ways to choose and order a pyramid: 3 options
• Number of ways to choose and order a cube: 4 options
• Number of ways to choose and order a sphere: 7 options

Thus, total ordered arrangements:

\[ 3 \times 4 \times 7 = 84 \]

However, since order preservation involves permutations across the chosen objects, the total is:

\[ 3 \times 4 \times 7 = 84 \]

In this case, because selection is ordered, the total number of arrangements is the same as the previous case, 84, considering the unique objects with different colors and shapes.

Additional Problems

#2: How many cards are in the deck in the game of Sets?

The game of Sets features cards with combinations of attributes: number of shapes (1, 2, 3), shape type (squiggle, diamond, oval), color (green, purple, red), and style (solid, shaded, outline). The total number of cards equals the product of possibilities for each attribute:

\[ 3 \text{ (number)} \times 3 \text{ (shape)} \times 3 \text{ (color)} \times 3 \text{ (style)} = 3^4 = 81 \]

#3: How many ways can the coach choose at least 2 good hitters out of 3 players selected from the bench?

The coach has 5 good hitters and 4 poor hitters, and he chooses 3 players in total. The number of ways to select at least 2 good hitters includes two scenarios:

• Exactly 2 good hitters and 1 poor hitter
• All 3 are good hitters

Calculations:

Number of ways to choose 2 good hitters and 1 poor hitter:

\[ \binom{5}{2} \times \binom{4}{1} = 10 \times 4 = 40 \]

Number of ways to choose all 3 good hitters:

\[ \binom{5}{3} = 10 \]

Adding these scenarios:

\[ 40 + 10 = 50 \]

Therefore, there are 50 ways to select at least 2 good hitters for the lineup.

References

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