Question

A telephone number has seven digits, no number starts with \(0 .\) In a city, how many different telephone numbers be formed using the digits 0 to \(6 ?\) (each digit can occur only once) (1) \(6 !\) (2) \(6.6 !\) (3) \(7 !\) (4) \(2 \cdot 7 !\)

Short answer

Answer Choices: 1) 6! 2) 6.6! 3) 7! 4) 7.7! Correct Answer: (2) 6.6!

Step-by-step solution

Determine the options for the first digit

Since a telephone number cannot start with 0, the first digit has 6 options (1, 2, 3, 4, 5, and 6).

Calculate the options for the remaining digits

After selecting the first digit, there will be 6 digits left to choose from (0 and the remaining 5 digits that weren't used as the first digit). So, for the second digit, there are 6 options available. For the third digit, there will be 5 options since two digits have already been chosen. Continuing this pattern, the number of options for the remaining digits are as follows: 6 options for the second digit, 5 options for the third digit, 4 options for the fourth digit, 3 options for the fifth digit, 2 options for the sixth digit, and 1 option for the seventh digit.

Calculate the total number of possible combinations

To find the total number of possible telephone numbers, multiply the number of options for each digit together: 6(options for first digit) x 6(options for second digit) x 5(options for third digit) x 4(options for fourth digit) x 3(options for fifth digit) x 2(options for sixth digit) x 1(option for seventh digit)

Simplify the expression

Simplify the expression, 6 x 6 x 5 x 4 x 3 x 2 x 1 which can be written as 6 x (6!).

Find the correct answer from given options

From the calculations, we determined that the total number of possible telephone numbers that can be formed is 6 x (6!). The correct answer is (2) 6.6!.

Key concepts

Permutation

Permutation is a fundamental concept in combinatorics. It refers to the different ways of arranging a set of items or numbers. In simpler terms, it’s about figuring out all possible orderings of a set of elements.

Imagine you are arranging books on a shelf. If you have five books, the total number of ways you can arrange them follows the principles of permutation. In permutations, the order really matters. For example, arranging [1, 2, 3] is different from arranging [3, 2, 1]. In this scenario, every different sequence is a unique permutation.

Permutations are used to solve problems where the arrangement or order of items is crucial, just as in the case of forming telephone numbers from digits.

Factorial notation

Factorial notation is a mathematical tool used to simplify the calculation of permutations and combinations. It is denoted by an exclamation point "!". For any positive integer n, the factorial n! is the product of all positive integers less than or equal to n. In mathematical terms, \[ n! = n \times (n-1) \times (n-2) \times \, ... \, \times 3 \times 2 \times 1 \]

Factorials play a crucial role in counting and arrangements. For example, if you have four objects, the factorial of 4, which is 4! = 4 x 3 x 2 x 1, quantifies all the possible ways to arrange those objects. In the original exercise, calculating the number of telephone numbers involves understanding that 6! represents all possible permutations of the numbers.

Telephone numbers

Telephone numbers are not only a means of communication but an excellent example of permutations in real life. In the problem we have, a city uses seven-digit numbers that can include any digits from 0 to 6, but cannot start with the digit 0. This constraint is crucial, as the first digit defines the number's validity for making calls.

When calculating possible phone numbers, only digits 1 to 6 can be considered for the first position, leading us to engage permutation principles to determine the overall arrangement possibilities. Constraints like starting conditions or rules on repetition affect the total number of combinations, demonstrating the direct application of combinatorics to common practical problems.

Digits arrangement

Digits arrangement is a practical application of permutation where you arrange selected digits in a sequence. The original problem deals with arranging seven digits for a phone number, ensuring that no digit repeats, and that the phone number doesn’t begin with 0. This limit makes sure that numbers are reliable and practical for use.

For a sequence of digits, we start by selecting the first digit from a pool of numbers, and then arrange the remaining digits. Each time we choose a new digit, the available pool decreases, following a patterned countdown. This ensures that all possible variations of the digits are considered in our calculations. These principles guide us in determining how many unique phone numbers are possible, reinforcing the concept of permutations and factorial notations.