Problem 1 :
A mobile phone has a passcode of 6 distinct digits. What is the maximum number of attempts one makes to retrieve the passcode?
Solution :
The passcode must be formed using the following digits
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Each digits must be different.
Total number of ways = 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5
= 151,200
Since we have to use distinct digits, we cannot choose the repeated numbers.
Hence the total number of ways of retrieving the passcode is 151200.
Problem 2 :
Given four flags of different colors, how many different signals can be generated if each signal requires the use of three flags, one below the other?
Solution :
1^{st} flag may be chosen out of 4 flags, 2^{nd} flag may be chosen out of 3 flags and 3^{rd} flag may be chosen out of 2 flags.
Total number of ways = 4 ⋅ 3 ⋅ 2
= 24 ways
Hence 24 different signals may be formed using 4 flags.
Problem 3 :
Four children are running a race.
(i) In how many ways can the first two places be filled?
(ii) In how many different ways could they finish the race?
Solution :
(i) Out of 4 children, any one may get first prize. Out of three children, any one may get the second prize.
Hence the total number of ways = 4 (3) = 12 ways.
(ii) Out of 4 ----> Any one may get 1^{st} prize
Out of 3 ----> Any one may get 2^{nd} prize
Out of 2 ----> Any one may get 3^{rd} prize
Remaining 1 will get fourth place.
Hence total number of ways = 4 ⋅ 3 ⋅ 2 ⋅ 1 = 24 ways
Problem 4 :
Count the number of three-digit numbers which can be formed from the digits 2, 4, 6, 8 if (i) repetitions of digits is allowed. (ii) repetitions of digits is not allowed
Solution :
Required three digit number
___ ___ ___
(i) repetitions of digits is allowed
Since repetition of digits is allowed, we have 4 options to fill each places.
Hence the numbers to be formed with the given digits are
= 4 ⋅ 4 ⋅ 4
= 64
(ii) repetitions of digits is not allowed
Hundred place :
We may use any of the digits (2, 4, 6, 8), so we have 4 options.
Tens place :
Repetition is not allowed.So, we have 3 options.
Unit place :
By excluding the number used in the hundreds and tens place, we have 2 options.
Hence total numbers to be formed = 4 ⋅ 3 ⋅ 2
= 24
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