Find the number of four digit even numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of those will be even?
Last updated date: 22nd Mar 2023
•
Total views: 304.8k
•
Views today: 6.83k
Answer
304.8k+ views
Hint: We use the principle of permutations and combinations to find the number of four digit even numbers using digits 1, 2, 3, 4 and 5. Thus, we use the fact that each digit has five possibilities and at the same time keeping in mind that we don’t repeat the digits. Further, to ensure that the number is even we ensure that in the unit’s place, the digits can only be 2 or 4.
Complete step-by-step answer:
First, since we have a constraint for the unit's place, we start by counting the combinations for the unit's place. Thus, we have only two possibilities – 2 or 4. Now, for the remaining places, we can start from any digit’s place to calculate the combinations. Suppose, we start with the tenths place, we now have five possibilities (1, 2, 3, 4 and 5) but we have to exclude one possibility since the unit’s place is already filled and no digit can be repeated. Thus, we would have four possibilities. Now, we move on to hundredths place, we would similarly have only 3 possibilities (since, now two places are filled and no digits can be repeated). Similarly, we would now have only two possibilities for the thousandths place. Thus, to calculate the total number of four digit even numbers, we multiply all possibilities to get the answer. Thus, we have, total possibilities-
2 $\times $ 4 $\times $ 3 $\times $ 2 = 48
Hence, there are 48 four digit even numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated.
Note: Generally, for questions regarding permutations and combinations involving calculations of number of digits, we start with the constraint portion of the problem. In this case, we start with the fact that we required the four digit number to be even and then proceed forward. This greatly helps us in simplifying the problem and further prevents from incorrectly counting the total numbers.
Complete step-by-step answer:
First, since we have a constraint for the unit's place, we start by counting the combinations for the unit's place. Thus, we have only two possibilities – 2 or 4. Now, for the remaining places, we can start from any digit’s place to calculate the combinations. Suppose, we start with the tenths place, we now have five possibilities (1, 2, 3, 4 and 5) but we have to exclude one possibility since the unit’s place is already filled and no digit can be repeated. Thus, we would have four possibilities. Now, we move on to hundredths place, we would similarly have only 3 possibilities (since, now two places are filled and no digits can be repeated). Similarly, we would now have only two possibilities for the thousandths place. Thus, to calculate the total number of four digit even numbers, we multiply all possibilities to get the answer. Thus, we have, total possibilities-
2 $\times $ 4 $\times $ 3 $\times $ 2 = 48
Hence, there are 48 four digit even numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated.
Note: Generally, for questions regarding permutations and combinations involving calculations of number of digits, we start with the constraint portion of the problem. In this case, we start with the fact that we required the four digit number to be even and then proceed forward. This greatly helps us in simplifying the problem and further prevents from incorrectly counting the total numbers.
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

The coordinates of the points A and B are a0 and a0 class 11 maths JEE_Main

Trending doubts
Write an application to the principal requesting five class 10 english CBSE

Tropic of Cancer passes through how many states? Name them.

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

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE

What is per capita income

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

A Short Paragraph on our Country India
