Notes

Problems & Videos

\(\textbf{1)}\) \( 5! \)

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The answer is \( 120 \)

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\(\,\,\,\,\,\,5!=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120\)

 

\(\textbf{2)}\) \( C(6,2)\) or \({}_{6}C_{2}\) or \(6 \choose 2\)

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The answer is \( 15 \)

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\(\,\,\,\,\,\,{}_{n}C_{r}=\displaystyle \frac{n!}{r!(n-r)!}\)
\(\,\,\,\,\,\,{}_{6}C_{2}=\displaystyle \frac{6!}{2!(6-2)!}\)
\(\,\,\,\,\,\,{}_{6}C_{2}=\displaystyle \frac{6!}{2!\cdot4!}\)
\(\,\,\,\,\,\,{}_{6}C_{2}=\displaystyle \frac{6 \cdot 5 \cdot 4!}{2 \cdot 1 \cdot 4!}\)
\(\,\,\,\,\,\,{}_{6}C_{2}=\displaystyle \frac{30}{2}\)
\(\,\,\,\,\,\,\)The answer is \( 15 \)

 

\(\textbf{3)}\) \( P(4,3) \) or \({}_{4}P_{3}\)

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The answer is \( P(4,3)=24 \)

 

\(\textbf{4)}\) \( P(50,48) \) or \({}_{50}P_{48}\)

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The answer is \( P(50,48)=1.52 \times 10^{64} \) (Huge number)

 

\(\textbf{5)}\) You have 10 books on your shelf, you need to bring 3 books on vacation. How many different ways can you select the books?

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The answer is \( C(10,3)=120 \)

 

\(\textbf{6)}\) You have 10 books in your bag you need to put on the shelf. How many different arrangements can you make on your shelf?

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The answer is \( P(10,10)=3,628,800 \)

 

\(\textbf{7)}\) A tub contains 5 blue, 2 green, and 7 red marbles. How many different ways can 2 blue, 1 green, and 4 red marbles be seleced?

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The answer is \( C(5,2) \cdot C(2,1) \cdot C(7,4)=700 \)

 

\(\textbf{8)}\) A box contains 8 blue marbles and 6 green marbles. You select 8 marbles. What is the probability that half are blue and half are green?

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The answer is \( \displaystyle\frac{{}_{8}C_{4} \cdot {}_{6}C_{4}}{{}_{14}C_{8}} \approx 0.3497\)

 

See Related Pages\(\)

\(\bullet\text{ Statistics Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Factorials}\)
\(\,\,\,\,\,\,\,\,5!= 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1…\)

 

In Summary

Combinations and permutations are two related concepts in mathematics that deal with the ways in which objects from a set can be selected or arranged.

A combination is a selection of items from a set, where the order of the items does not matter. For example, if you have a set of three letters {A, B, C}, the possible combinations of two letters are {A, B}, {A, C}, and {B, C}. There are three possible combinations of two letters from this set. Factorials are used in the formula for a combinations.

\({}_{n}C_{r}=\displaystyle \frac{n!}{r!(n-r)!}\)

A permutation, on the other hand, is an arrangement of items from a set, where the order of the items does matter. For example, if you have the same set of three letters {A, B, C}, the possible permutations of two letters are {A, B}, {B, A}, {A, C}, {C, A}, {B, C}, and {C, B}. There are six possible permutations of two letters from this set. Factorials are also used in the formula for a permutations.

\({}_{n}P_{r}=\displaystyle \frac{n!}{(n-r)!}\)

Combinations and permutations are often used in probability and statistics, as well as in other areas of mathematics. They are useful for computing the number of possible ways in which objects from a set can be selected or arranged.