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How many even numbers, larger than 400, with 3 different digits, can be formed using digits 1, 2, 3, 4, 5, 6, 7, 8?
A. 102
B. 620
C. 640
D. 2048
E. 2520
I am struggling to understand why 102 is the correct answer as my approach is; 5 options for the first digit, 7 for the second and from there it is already wrong because there is no integer to multiply 5*7 that result in 102. Thank you in advance
P.D.: I am sorry for any possible mistake as I am new in the forum. I could not find the right topic for the problem because I have it as a counting methode. The source is from an online platform provided by my Academy.
A. 102
B. 620
C. 640
D. 2048
E. 2520
I am struggling to understand why 102 is the correct answer as my approach is; 5 options for the first digit, 7 for the second and from there it is already wrong because there is no integer to multiply 5*7 that result in 102. Thank you in advance
P.D.: I am sorry for any possible mistake as I am new in the forum. I could not find the right topic for the problem because I have it as a counting methode. The source is from an online platform provided by my Academy.
alfonspm
1. If the hundreds digit is even (so 4, 6, or 8):
Number of options for the hundreds digit: 3;
Number of options for the units digit: 3 (any from 2, 4, 6, or 8 but one used for the hundreds);
Number of options for the tens digit: 6.
Total = 3*3*6 = 54.
2. If the hundreds digit is odd (so 5, or 7):
Number of options for the hundreds digit: 2;
Number of options for the units digit: 4 (any from 2, 4, 6, or 8);
Number of options for the tens digit: 6.
Total = 2*4*6 = 48.
GRAND TOTAL = 54 + 48 = 102.
Answer: A.
alfonspm
The Category for such kind of questions in Combinations.
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Sorry I don't understand why you split the hundreds digits into even or odds. If the number must be even the only digit you should split is the units, as are the ones that are gonna tell you whether is even or odd.
alfonspm
If you don't split you won't get correct answer. If you check the solution you'll see that the number of options of the hundreds digit affects the number of options of the units digit.
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alfonspm
Look at it this way
Case 1: If Unit digit is taken by digit 2 then hundreds digit has 5 options and tens digit has 6 options i.e. 1*5*6 = 30 numbers
Case 2: If Unit digit is taken by digit 4 or 6 or 8 then hundreds digit has 4 options and tens digit has 6 options i.e. 3*4*6 = 72 numbers
Total Possibilities = 30+72 = 102
Bunuel has tried it in different ways because when hundreds digit is 4 or 6 or 8 then unit digit has 3 possibilities but when Hundreds digit is 5 or 6 then Unit digits have 4 possibilities so the cases can not be clubbed together hence they need to be solved as separate cases
I hope this helps!!!
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It does thank you very much!
To get # of numbers greater than 400 and even, first digit can be any number from 4 to 8 while last number should be an even number.
if the last number is 2, then we have 1 way to fill the last digit; 100's place can be filled by 4,5,6,7,8(total 5) and 10th place can be filled with any number from 1 to 8, since two numbers are already being filled in last digit and 100th place, we can take only 6 digits, thus total number of numbers that end with two is 1*5*6=30
if the last number is 4,6,8, then we have 3 ways to fill the last digit; 100's place can be filled by 5,7, and any two of {4,6,8} (total 4), as the other number is filled in last digit earlier. and 10th place can be filled with any number from 1 to 8, since two numbers are already being filled in last digit and 100th place, we can take only 6 digits, thus total number of numbers that end with either [4,6,8] is 3*4*6 = 72 numbers
Total number of numbers = 30+72 = 102
if the last number is 2, then we have 1 way to fill the last digit; 100's place can be filled by 4,5,6,7,8(total 5) and 10th place can be filled with any number from 1 to 8, since two numbers are already being filled in last digit and 100th place, we can take only 6 digits, thus total number of numbers that end with two is 1*5*6=30
if the last number is 4,6,8, then we have 3 ways to fill the last digit; 100's place can be filled by 5,7, and any two of {4,6,8} (total 4), as the other number is filled in last digit earlier. and 10th place can be filled with any number from 1 to 8, since two numbers are already being filled in last digit and 100th place, we can take only 6 digits, thus total number of numbers that end with either [4,6,8] is 3*4*6 = 72 numbers
Total number of numbers = 30+72 = 102
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