# How many $5$ digit telephone numbers can be constructed using the digits $0$ to $9$ if each number starts with $67$ and no digit appears more than once?

Answer

Verified

381k+ views

Hint: From $0$ to $9$, two digits are already fixed for first and second position. We have to fill the remaining three positions with the remaining 8 digits.

According to the question, $5$ digit telephone numbers are to be formed using digits $0$ to $9$ keeping in mind that no digit is repeated. And the number must start with $67$.

So out of the five places in the number, first two places are already taken by $67$. So we are left with only $3$ other places which are to be filled.

Further, from $0$ to $9$ there are $10$ digits. Out of them, digit $6$ and digit $7$ is already taken for first and second place respectively. So, now we are left with only $8$ more digits (because we have to avoid repetition of digits also).

Let’s suppose we are filling third place first. So, we have $8$ different digits for this place to be filled with and this can be done in $8$ different ways.

$\therefore $ The number of ways of filling third place is $8$.

Now, we are left with only $7$ more digits. So, the fourth place can be filled in $7$ different ways.

$\therefore $ The number of ways of filling fourth place is $7$.

For the last place, we have $6$ remaining digits. So, the last place can be filled in $6$ different ways.

$\therefore $ The number of ways of filling last place is $6$.

Therefore, by multiplication principle, the required number of ways in which five digit telephone numbers can be constructed is $8 \times 7 \times 6 = 336$.

Note: According to multiplication principle, if one event can occur in $m$ ways and a second event can occur in $n$ ways after the first event has occurred, then the two events can occur in $m \times n$ ways. This is also known as the Fundamental Counting Principle.

According to the question, $5$ digit telephone numbers are to be formed using digits $0$ to $9$ keeping in mind that no digit is repeated. And the number must start with $67$.

So out of the five places in the number, first two places are already taken by $67$. So we are left with only $3$ other places which are to be filled.

Further, from $0$ to $9$ there are $10$ digits. Out of them, digit $6$ and digit $7$ is already taken for first and second place respectively. So, now we are left with only $8$ more digits (because we have to avoid repetition of digits also).

Let’s suppose we are filling third place first. So, we have $8$ different digits for this place to be filled with and this can be done in $8$ different ways.

$\therefore $ The number of ways of filling third place is $8$.

Now, we are left with only $7$ more digits. So, the fourth place can be filled in $7$ different ways.

$\therefore $ The number of ways of filling fourth place is $7$.

For the last place, we have $6$ remaining digits. So, the last place can be filled in $6$ different ways.

$\therefore $ The number of ways of filling last place is $6$.

Therefore, by multiplication principle, the required number of ways in which five digit telephone numbers can be constructed is $8 \times 7 \times 6 = 336$.

Note: According to multiplication principle, if one event can occur in $m$ ways and a second event can occur in $n$ ways after the first event has occurred, then the two events can occur in $m \times n$ ways. This is also known as the Fundamental Counting Principle.

Recently Updated Pages

Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

Scroll valve is present in a Respiratory system of class 11 biology CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

is known as the Land of the Rising Sun A Japan B Norway class 8 social science CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE