SOLUTION: The total number of natural number of six digit thats can be made with digits 1,2,3,and 4 if all digits have to appear in the same number at least once is.

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Question 1135810: The total number of natural number of six digit thats can be made with digits 1,2,3,and 4 if all digits have to appear in the same number at least once is.
Answer by greenestamps(13215) About Me  (Show Source):
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There are two kinds of 6-digit numbers that contain only digits 1, 2, 3, and 4 with each of those digits occurring at least once:

(1) 3 of one of the digits and 1 each of the other three

You can choose any one of the 4 digits to be the one that is used 3 times: C%284%2C1%29+=+4
The number of different arrangements of the 6 digits is 6%21%2F%28%283%21%29%281%21%29%281%21%29%281%21%29%29+=++120

Total number of 6-digit numbers of this type: 4*120 = 480

(2) 2 each of 2 of the digits and 1 each of the other two

You can choose any 2 of the 4 digits to be the ones that are used twice: C%284%2C2%29+=+6
The number of different arrangements of the 6 digits is6%21%2F%28%282%21%29%282%21%29%281%21%29%281%21%29%29+=+180

Total number of 6-digit numbers of this type: 6*180 = 1080

Total number of 6-digit numbers that contain only digits 1, 2, 3, 4 with each being used at least once: 480+1080 = 1560