Question

Form 2 digit number using 0,1,2,3,4,5 without repeating the digits. Write the sample space S, number of sample points n(S), U, n(U) for U is the event that the number so formed is divisible by 5.

Answer 1

Hint:Here first we will find all the two digit numbers which can be formed using the given digits and write the sample space and also write the number of elements in sample space.Then we will list all the numbers which are divisible by 5 from the numbers listed in sample space S and write then in U and also write the number of elements in U so formed.A number is divisible by 5 if its unit’s place contains 0 or 5.

Complete step-by-step answer:
The given digits are:-
0,1,2,3,4,5
Now since we know that no two digit number can start with 0 hence we will first list all the numbers starting with 1:
10, 11, 12,13,14,15
Now, we will list all the numbers starting with 2:-
20, 21, 22,23,24,25
Now, we will list all the numbers starting with 3:-
30, 31, 32,33,34,35
Now, we will list all the numbers starting with 4:-
40, 41, 42,43,44,45
Now, we will list all the numbers starting with 5:-
50, 51, 52,53,54,55
Therefore the sample space S is given by:-
\[S = \left\{ {10,{\text{ }}11,{\text{ }}12,13,14,15,20,{\text{ }}21,{\text{ }}22,23,24,25,30,{\text{ }}31,{\text{ }}32,33,34,35,40,{\text{ }}41,{\text{ }}42,43,44,45,50,{\text{ }}51,{\text{ }}52,53,54,55} \right\}\]
Now counting the number of elements in S we get:-
\[n\left( S \right) = 25\]
Now since we know that a number is divisible by 5 if its unit’s place contains 0 or 5.
Hence listing such numbers from S which have 0 or 5 at unit’s place we get:-
10,15,20,25,30,35,35,40,45,50,55
Since the U is the event that it contains the number which is divisible by 5
Hence, U is given by:-
\[U = \left\{ {10,15,20,25,30,35,35,40,45,50,55} \right\}\]
Now counting the number of elements in U we get:-
\[n\left( U \right) = 11\]
Therefore,
Sample space S is:
\[S = \left\{ {10,{\text{ }}11,{\text{ }}12,13,14,15,20,{\text{ }}21,{\text{ }}22,23,24,25,30,{\text{ }}31,{\text{ }}32,33,34,35,40,{\text{ }}41,{\text{ }}42,43,44,45,50,{\text{ }}51,{\text{ }}52,53,54,55} \right\}\]
Number of elements in S are:-
\[n\left( S \right) = 25\]
The event U is:-
\[U = \left\{ {10,15,20,25,30,35,35,40,45,50,55} \right\}\]
Number of elements in U are:-
\[n\left( U \right) = 11\].

Note:Students should note that a number is divisible by 5 if and only if its unit’s place contains 0 or 5, all other numbers are not divisible by 5.Also in such questions, it is advised to write the whole sample space first i.e. writing all the possible outcomes according to the given condition.