Outside a home, there is a 7-key keypad numbered 1 through 7. The correct five-digit code will open the garage door. The numbers can be repeated in the code. (a) How many codes are possible? (b) What is the probability of entering the correct code on the first try, assuming that the owner doesn't remember the code?

(a) How many codes are possible is 16,807

(b) The probability of entering the correct code on the first  is 5.95%

(a) How many codes are possible

Since we have 7 number that can placed in 5 places which means the possible code will be calculated as:

Possible Code =7×7×7×7×7 or 7^5

Possible Code =16,807

(b) The probability of entering the correct code on the first  try will be:

Since only 1 correct code is correct

Hence:

Probability=1/16,807

Probability=5.949%

Probability=5.95% (Approximately)

Inconclusion (a) How many codes are possible is 16,807

(b) The probability of entering the correct code on the first  is 5.95%

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a: 16807

b: 0.0000595

Step-by-step explanation:

(a): We have 7 possibilities of numbers, and we need to choose 5 numbers that can repeat. So, for each of the 5 numbers, we have 7 possibilities. To determine how many codes are possible, we just need to multiply the number of possibilities of each of the 5 numbers, so we have 7x7x7x7x7, or 7^5, which is 16807 possibilities.

(b): to put the correct code in the first try, we need to find the exact combination (just one combination) among these 16807 possibilities, so the probability is 1/16807, which is 0.0000595, or 0.00595%.

answer: (x+2) (x+3) are the binomials for your question : )