In Iceland, phone numbers consist of seven digits, and phone numbers are allowed to start with 0. For example, three possible Icelandic phone numbers are 031-4159, 555-1234, and 867-5309. How many possible Icelandic phone numbers contain at least five 5’s?
1- For Exactly 5 "fives", we have the follwing:
The 5 "fives" included in the 7-digit telephone number can be ordered in: 7 C 5 = 21 ways. The remaining 2 spots can be occupied by any of the remaining 9 digits, since all digits can be repeated. So that is: 9 x 9 = 81.[We cannot have 5 included in there because we are counting EXACTLY 5 "fives".
Therefore, we have: [7 C 5] * 9^2 =1701 - Telephone numbers with exactly 5 "fives"
2 - Now will calculate exactly 6 "fives". It is basically the same calculation as above: [7 C 6] * 9 = 63 - Telephone numbers with exactly 6 "fives".
3 - Finally, will calculate exactly 7 "fives". That comes to: [7 C 7] = 1 - Telephone number with 7 "fives"
4 - Summing them up, we have: 1701 + 63 +1 =1765 - Telephone numbers that have AT LEAST 5 "fives".
