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29 Mar 2020, 21:59
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In how many ways we can create 4 digits even numbers without repetition?
30 Mar 2020, 23:42
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Hello Bijay,
Although this question does not look exactly like a GMAT question, it nevertheless serves the purpose of providing much needed practice on permutations with numbers and digits.
Since the question does not tell us anything about what digits can be used, I’m assuming that we can use all the digits of the decimal system i.e. 0 to 9.
Since we need even numbers without repetition, we need to work with the constraint that the units digit of the numbers should be 0/2/4/6/8 and once we put any of these digits in the units place, they cannot be repeated.
As such, the best method to solve this question is to take cases. Let’s consider two cases.
Case 1: The 4 digit number ends with 0.
In this case, the thousands digit can be filled in 9 ways since any of the digits amongst 1 to 9 can come in this place. Since repetition is not allowed, the hundreds digit can be filled in 8 ways and the tens digit can be filled in 7 ways.
Since we need to fill all the digits simultaneously, the number of ways of filling up the thousands and the hundreds and the tens digits = 9 x 8 x 7 = 504 ways.
Case 2: The 4 digit number ends with 2.
In this case, the thousands digit can be filled in 8 ways – that’s because we have already used up 2 and cannot repeat it; we cannot use 0 in this place since that would then make the number a 3-digit number.
The hundreds digit can also be filled in 8 ways – 0 can be filled in this place.
The tens digit can be filled in 7 ways.
Therefore, total number of 4-digit numbers with distinct digits which end with the digit 2 = 8*8*7 = 448.
Note that numbers ending with 4, 6 and 8 will also behave exactly like numbers ending with 2. Therefore, total number of 4-digit numbers with distinct digits which end with 2/4/6/8 = 448 * 4 = 1792.
Adding this with the even numbers ending with the digit 0, we have total 4-digit numbers = 1792 + 504 = 2296.
Note that you cannot directly use the Fundamental Principle of Counting in questions like these because of the fact that certain cases behave differently compared to certain other cases. So, creating separate cases is the best way to solve these type of questions with additional constraints.
Hope that helps!
Although this question does not look exactly like a GMAT question, it nevertheless serves the purpose of providing much needed practice on permutations with numbers and digits.
Since the question does not tell us anything about what digits can be used, I’m assuming that we can use all the digits of the decimal system i.e. 0 to 9.
Since we need even numbers without repetition, we need to work with the constraint that the units digit of the numbers should be 0/2/4/6/8 and once we put any of these digits in the units place, they cannot be repeated.
As such, the best method to solve this question is to take cases. Let’s consider two cases.
Case 1: The 4 digit number ends with 0.
In this case, the thousands digit can be filled in 9 ways since any of the digits amongst 1 to 9 can come in this place. Since repetition is not allowed, the hundreds digit can be filled in 8 ways and the tens digit can be filled in 7 ways.
Since we need to fill all the digits simultaneously, the number of ways of filling up the thousands and the hundreds and the tens digits = 9 x 8 x 7 = 504 ways.
Case 2: The 4 digit number ends with 2.
In this case, the thousands digit can be filled in 8 ways – that’s because we have already used up 2 and cannot repeat it; we cannot use 0 in this place since that would then make the number a 3-digit number.
The hundreds digit can also be filled in 8 ways – 0 can be filled in this place.
The tens digit can be filled in 7 ways.
Therefore, total number of 4-digit numbers with distinct digits which end with the digit 2 = 8*8*7 = 448.
Note that numbers ending with 4, 6 and 8 will also behave exactly like numbers ending with 2. Therefore, total number of 4-digit numbers with distinct digits which end with 2/4/6/8 = 448 * 4 = 1792.
Adding this with the even numbers ending with the digit 0, we have total 4-digit numbers = 1792 + 504 = 2296.
Note that you cannot directly use the Fundamental Principle of Counting in questions like these because of the fact that certain cases behave differently compared to certain other cases. So, creating separate cases is the best way to solve these type of questions with additional constraints.
Hope that helps!
