How many eight-digit serial numbers contain three 9\'s, two 5\'s, two 4\'s, and

Question

How many eight-digit serial numbers contain three 9's, two 5's, two 4's, and one 0? How many eight-digit serial numbers contain three of one digit, two of another digit, two of yet another digit, and one of still another digit? (A total of four different digits must be involved, in the given quantities.) 3. 4. . How many eight-digit serial numbers contain either five even digits (and three odd digits) or six even digits (and two odd digits)? Since there are usually multiple runs in which all possibilities for the eight digits are used, it is reasonable to assume that any one eight-digit serial number is as likely to appear on your dollar bill as any other eight-digit serial number. Use this assumption as you answer the final question. What is the probability that an eight-digit serial number will contain exactly five or six even digits? 6.

Explanation / Answer

(4)

Since you have highlighted the fourth part, I will answer that only.

There are total 10 digits available to construct a serial number of a length of eight digits.

Total possible number of ways of forming a serial number without any restrictions = 10^8

Now we consider the possible formations with the given conditions:

First we must select only 4 digits out of the total 10 digits.

The number of ways of doing this are: 10C4

Next we have to choose 2 out of those 4 selected, which will occur twice. And then out of the remaining two, we have to choose which one should occur thrice.

Total number of ways of making these selections are: 4C2*2

Now after we finally make these selections, we then have to make unique arrangements to get different serial numbers.

Total unique numbers that can be made after the above selections have been made = 8!/(3!*2!*2!)

So, total number of ways of constructing these 8-digit serial numbers = 10C4*4C2*2*8!/(3!*2!*2!) = 4233600

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