Question

A keypad code consists of a string of symbols of length 6 from set {1, 2, 3, 4, 5, 6, 7, 8, 9}. Note that the digit 0 is not allowed.

A) how many keypad codes of length 6 have at least one repeated symbol?

B) how many such keypad codes use the symbol 6 at least once and use the symbol 7 at least once?

Answer 1

A) Total number of strings of length 6 = 9⁶

= 531,441

Number of strings of length 6 without repeating = 9P6

= 9!/(9-6)!

= 60,480

Number of keypad codes of length 6 that have at least one repeated symbol = Total number of strings of length 6 - Number of strings of length 6 without repeating

= 531,441 - 60,480

= 470,961

B) Number of keypad codes that does use the symbol 6 at least once and use the symbol 7 at least once = Number of keypad codes that does use the symbol 6 + Number of keypad codes that does use the symbol 7 + Number of keypad codes that does use both 6 and 7 - Total number of keypad codes

= 8⁶ + 8⁶ + 7⁶ - 9⁶

= 110,496