How many strings of four decimal digits (Note there are 10 possible digits and a string can be of the form 0014 etc., i.e., can start with zeros.)
(a) have exactly three digits which are 9s?
Think of 3 9's to which we have to insert a fourth digit x(which is not a nine).
This can be in the four following ways.
X999 or 9X99 or 99X9 or 999X
In each case X can represent 9 possible digits including 0 but excluding 9.
So the number of strings is 4*9=36.
To determine the number of strings with exactly three digits which are 9s, we can consider the positions of the 9s in the string.
Since there are four positions in the string, we can choose any three of them to be filled with 9s. The remaining position can be filled with any of the remaining 10 digits (0-9).
To calculate this, we can use the combination formula. The number of ways to choose three positions out of four is given by:
C(4, 3) = 4! / (3! * (4-3)!) = 4
Once we have chosen the positions for the 9s, each remaining position can be filled with any of the 10 digits.
Therefore, the total number of strings with exactly three digits which are 9s is:
4 * 10 = 40
So, there are 40 strings with exactly three digits which are 9s.
To find the number of strings of four decimal digits with exactly three 9s, we can break down the problem into steps:
Step 1: Determine the possible positions for the three 9s. Since there are four positions in total and exactly three of them need to be occupied by 9s, we can choose the positions in the following ways:
- The three 9s can be in positions 1, 2, and 3.
- The three 9s can be in positions 1, 3, and 4.
- The three 9s can be in positions 2, 3, and 4.
Step 2: For each of the possible combinations of three 9s, we need to fill the remaining position(s) with digits other than 9. Since there are 10 possible digits (0-9) and we cannot use 9 in the remaining position(s), we have 9 choices for each remaining position.
Step 3: Compute the total number of strings by summing up the possibilities from each step.
Let's calculate the total number of strings:
For the first combination (positions 1, 2, and 3):
- The first position must be filled with a 9, leaving two 9s remaining.
- The second and third positions can be filled with any non-9 digit, giving us 9 choices for each position.
So, the total number of strings for this combination is: 1 * 9 * 9 = 81.
For the second combination (positions 1, 3, and 4):
- The first position must be filled with a 9, leaving two 9s remaining.
- The second and fourth positions can be filled with any non-9 digit, giving us 9 choices for each position.
So, the total number of strings for this combination is: 1 * 9 * 9 = 81.
For the third combination (positions 2, 3, and 4):
- The second position must be filled with a 9, leaving two 9s remaining.
- The third and fourth positions can be filled with any non-9 digit, giving us 9 choices for each position.
So, the total number of strings for this combination is: 1 * 9 * 9 = 81.
Finally, we sum up the possibilities from each combination: 81 + 81 + 81 = 243.
Therefore, there are 243 strings of four decimal digits with exactly three digits being 9s.