Suppose a four-character PIN must be created using only the digits 1–7. (a) How
ID: 3141052 • Letter: S
Question
Suppose a four-character PIN must be created using only the digits 1–7.
(a) How many such PINs are there in total?
(b) How many such PINs are composed of four distinct digits? For example, 2417.
(c) How many such PINs are composed of only three distinct digits? For example, 1337.
(d) How many such PINs have the property that their digits sum to 6 or less? For example, 3111.
(e) How many such PINs start with a 7? For example, 7334.
(f) How many such PINs start with a 7 and end with an odd digit? For example, 7333.
(g) How many such PINs have two 1s in the middle? For example, 2114.
(h) How many such PINs have exactly two 1s? For example, 1712.
(i) How many such PINs have at least two 1s? For example, 1211.
(j) How many such PINs do not contain a 6 anywhere? For example, 1234.
Explanation / Answer
(According to Chegg policy, only four subquestions will be answered. Please post the remaining in another question)
(a) The first character could be any of the 7 digits.
The second character could be any of the 7 digits.
The third character could be any of the 7 digits.
The fourth character could be any of the 7 digits.
=> Total number of PINs = 7*7*7*7 = 74 = 2401
(b) The first character could be any of the 7 digits.
The second character could be any of the 6 digits as one is already taken.
The third character could be any of the 5 remaining digits.
The fourth character could be any of the 4 remaining digits.
=> Total number of PINs = 7*6*5*4 = 840
(c) Three digits are distinct and one digit repeats.
Total number of PINs = 840 / 2! = 420.
(d) If 4,5,6 or 7 are chosen, the sum will be greater than 6.
If 3 is chosen, the remaining three digits must all be 1s.
This can be done in 4 ways (the digit 3 can be any of the 4 characters)
If 2 is chosen, we have two choices:
(1) 2,2,1,1.
The number of such PINs = 840/(2!2!) = 210
(2) 2,1,1,1
The number of such PINs = 840/(3!) = 140
Only 1s can generate 1 pin i.e 1111.
So the total number of PINs = 4 + 210 + 140 + 1 = 355
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