# Combinations/permutations help

• roam
In summary, the conversation discusses how many natural numbers less than 100 contain a 3. The attempt at a solution involves counting the numbers of the form xy where x or y is a 3. However, it is easy to overcount and there are no easy systematic ways of finding the answer. The only number that can be counted is 33.
roam

## Homework Statement

How many (natural) numbers less than 100 contain a 3? (Note: 13, 35 and 73 all
contain a 3 but 42, 65 and 88 do not).

## The Attempt at a Solution

Of course I know that the numbers containing a 3 including 10 numbers starting with a 3 (30, . . . 39),and 10 numbers ending in a 3 (3, 13, . . . , 93), with 33 being counted twice, so a total of 19 numbers. I've found this by counting. But is there a quick systematic way of obtaining this answer using combinations/permutations etc? Unfortunently my knowledge of combinatorics is very poor, so I appreciate any help.

Between 10 to 100 there are 98 2-digit numbers that can possibly contain a 3 in the 1's or 10's positions... I'm stuck here.

Not every combinatorics problem involves permutations or combinations.

Think of it this way: You want to count the numbers of the form xy where x or y is a 3.

It would be easy to overcount, so be careful.

awkward said:
Not every combinatorics problem involves permutations or combinations.

Think of it this way: You want to count the numbers of the form xy where x or y is a 3.

It would be easy to overcount, so be careful.

The only count would be 33. So other than counting, there are no easy systematic ways of doing this?

How many numbers are there where x is a 3?

How many numbers are there where y is a 3?

## 1. What is the difference between combinations and permutations?

Combinations and permutations are both ways of arranging objects, but they differ in whether or not the order matters. Permutations refer to the number of ways objects can be arranged in a specific order, while combinations refer to the number of ways objects can be selected without regard to order.

## 2. How do I calculate the number of combinations or permutations?

The formula for combinations is nCr = n!/r!(n-r)!, where n is the total number of objects and r is the number of objects being selected. The formula for permutations is nPr = n!/(n-r)!, where n is the total number of objects and r is the number of objects being selected. Alternatively, you can use a combination or permutation calculator to quickly get the answer.

## 3. Can you give an example of combinations and permutations in real life?

Combinations and permutations can be seen in various aspects of daily life, such as when choosing lottery numbers, creating a password with certain characters, or arranging guests at a dinner party. In each of these cases, the order of the objects (numbers, characters, guests) matters, making them examples of permutations. On the other hand, selecting a group of students to work on a project or choosing a pizza with different toppings can be seen as examples of combinations, as the order of selection does not affect the outcome.

## 4. Are there any shortcuts or tricks for solving combinations and permutations?

Yes, there are a few shortcuts and tricks that can help you solve combinations and permutations quickly. For example, if you need to find the number of permutations of a set of objects where some are identical, you can use the formula n!/a!b!... where a, b, etc. represent the number of identical objects. Another trick is to use the "n choose r" button on a calculator to quickly calculate combinations.

## 5. How can I use combinations and permutations in my research or experiments?

Combinations and permutations can be useful in various scientific fields, such as genetics, chemistry, and statistics. For example, in genetics, combinations and permutations can be used to calculate the probability of certain traits being inherited. In chemistry, they can be used to determine the number of possible molecular structures. In statistics, they can be used to analyze and interpret data. Knowing how to calculate combinations and permutations can help in designing experiments and understanding the results.

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