1. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
(i) repetition of the digits is allowed? (ii) repetition of the digits is not allowed?
(i) repetition of the digits is allowed
Given that, 3 digit number must be formed from 5 digits, repetition allowed.
Units place can be filled in 5 ways.
Tens place can be filed in 5 ways.
Hundreds place can be filled in 5 ways.
Therefore, number of ways in which 3-digit number can be formed = 5*5*5=125.
(ii) repetition of the digits is not allowed?
Given that, 3 digit number must be formed from 5 digits, repetition not allowed.
Units place can be filled in 5 ways.
Tens place can be filed in 4 ways.
Hundreds place can be filled in 3 ways.
Therefore, number of ways in which 3-digit number can be formed = 5*4*3=60.
2. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
Given that, 3 digit even number must be formed from 6 digits, repetition allowed.
Units place can be filled with only 3 digits 2,4,6 since we need a even number.
Units place can be filled in 3 ways.
Tens place can be filled in 6 ways.
Hundreds place can be filled in 6 ways.
Therefore, number of ways in which 3-digit even number can be formed = 3*6*6=108.
3. How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Given that, 4 letter code must be formed using 10 letter, repetition not allowed.
First letter of code can be chosen in 10 ways.
Second letter of code can be chosen in 9 ways.
Third letter of code can be chosen in 8 ways.
Fourth letter of code can be chosen in 7 ways.
Therefore, the total number of 4 letter code=10*9*8*7=5040.
4. How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Given that, 5-digit number must be formed using 10 digits (i.e. 0 to 9), repetition not allowed.
Also, each number starts with 67 i.e. first 2 digits are 6,7. Remaining 3-digits must be formed using remaining 8 digits (0,1,2,3,4,5,8,9).
Units place can be filled in 8 ways.
Tens place can be filled in 7 ways.
Hundreds place can be filled in 6 ways.
Therefore, number of ways in which 5-digit number can be formed = 8*7*6=336.
5. A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
Given that, a coin is tossed 3 times.
When a coin is tossed, there are 2 possible outcomes Head(H) or Tail(T).
In each toss, there are 2 possible outcomes (H/T)
Therefore, number of possible outcomes if coin is tossed 3 times= 2*2*2=8.
6. Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?
Given that, Each signal requires 2 flags.
So, 2 vacant places must be filled using 5 flags.
Upper flag is filled in 5 ways.
Lower flag is filled in 4 ways.
Therefore, number of signals generated= 5*4=20.
(i) repetition of the digits is allowed? (ii) repetition of the digits is not allowed?
(i) repetition of the digits is allowed
Given that, 3 digit number must be formed from 5 digits, repetition allowed.
Units place can be filled in 5 ways.
Tens place can be filed in 5 ways.
Hundreds place can be filled in 5 ways.
Therefore, number of ways in which 3-digit number can be formed = 5*5*5=125.
(ii) repetition of the digits is not allowed?
Given that, 3 digit number must be formed from 5 digits, repetition not allowed.
Units place can be filled in 5 ways.
Tens place can be filed in 4 ways.
Hundreds place can be filled in 3 ways.
Therefore, number of ways in which 3-digit number can be formed = 5*4*3=60.
2. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
Given that, 3 digit even number must be formed from 6 digits, repetition allowed.
Units place can be filled with only 3 digits 2,4,6 since we need a even number.
Units place can be filled in 3 ways.
Tens place can be filled in 6 ways.
Hundreds place can be filled in 6 ways.
Therefore, number of ways in which 3-digit even number can be formed = 3*6*6=108.
3. How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Given that, 4 letter code must be formed using 10 letter, repetition not allowed.
First letter of code can be chosen in 10 ways.
Second letter of code can be chosen in 9 ways.
Third letter of code can be chosen in 8 ways.
Fourth letter of code can be chosen in 7 ways.
Therefore, the total number of 4 letter code=10*9*8*7=5040.
4. How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Given that, 5-digit number must be formed using 10 digits (i.e. 0 to 9), repetition not allowed.
Also, each number starts with 67 i.e. first 2 digits are 6,7. Remaining 3-digits must be formed using remaining 8 digits (0,1,2,3,4,5,8,9).
Units place can be filled in 8 ways.
Tens place can be filled in 7 ways.
Hundreds place can be filled in 6 ways.
Therefore, number of ways in which 5-digit number can be formed = 8*7*6=336.
5. A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
Given that, a coin is tossed 3 times.
When a coin is tossed, there are 2 possible outcomes Head(H) or Tail(T).
In each toss, there are 2 possible outcomes (H/T)
Therefore, number of possible outcomes if coin is tossed 3 times= 2*2*2=8.
6. Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?
Given that, Each signal requires 2 flags.
So, 2 vacant places must be filled using 5 flags.
Upper flag is filled in 5 ways.
Lower flag is filled in 4 ways.
Therefore, number of signals generated= 5*4=20.
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