Question

A garage has a letter-keypad lock with the ten letters A,B,C,D,E,F,G,H,I,J. The keypad can be programmed with a 4 or 6-letter code to open the garage. 1.Suppose each letter can be used more than once, how many 4-letter codes are possible if:

a) there are no restrictions?
b) the letters "A" and "B" must be used?
c) the letter "C" cannot be used?
d) the same letter is repeated four times?

Answer 1

The number of 4-letter codes from a 10-letter keypad varies based on specific restrictions: 10,000 with no restrictions, 2,400 requiring 'A' and 'B', 6,561 excluding 'C', and 10 when any letter is repeated four times.

Step-by-step explanation:

Combinations of Letter-Key Pad Codes

The subject in question involves calculating the number of possible combinations of a 4-letter code from a 10-letter keypad, where each letter can be used more than once. Here are the solutions to the provided scenarios:

a) No restrictions

For a 4-letter code with no restrictions, each position can be filled by any of the 10 letters. The total number of combinations is 10×10×10×10 which equals 10,000 possible 4-letter codes.

b) Letters "A" and "B" must be used

If "A" and "B" must be used once each in a 4-letter code, they can be arranged in 4! (4 factorial) ways, which equates to 24 arrangements. The remaining two positions can be filled by any of the 10 letters, so the total combinations are 24×10×10 totaling 2,400 possible codes.

c) The letter "C" cannot be used

Excluding the letter "C", we have 9 options for each position. The total number of combinations is 9×9×9×9 which equals 6,561 possible 4-letter codes.

d) The same letter repeated four times

For a 4-letter code where the same letter is repeated four times, there are just as many combinations as there are available letters. Hence, there are 10 possible codes, one for each letter.