Let S be the set of all passwords which are six to eight

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Question 1:- Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is \alpha× 5^6, then α is equal to

Solution:- Number of password using {1, 2, 3, 4, 5} and {A, B, C, D, E} having 6 character = 10^6

Number of password using {1, 2, 3, 4, 5} and {A, B, C, D, E} having 7 character = 10^7

Number of password using {1, 2, 3, 4, 5} and {A, B, C, D, E} having 8 character = 10^8

Total  password = 10^6 + 10^7 + 10^8

Number of password using {1, 2, 3, 4, 5} having 6 character =  5^6

Number of password using {1, 2, 3, 4, 5} having 7 character =  5^7

Number of password using {1, 2, 3, 4, 5} having 8 character =  5^8

Number of password with no number = 5^6 + 5^7 + 5^8[/latex]

Total number of password with at least one number =(10^6 + 10^7 + 10^8)-(5^6 + 5^7 + 5^8)

= 10^6(1 + 10 + 100)-5^6(1 + 5 + 25)

= 10^6\times 111 - 5^6\times 31

= 5^6( 64\times 111 - 31)

= 5^6( 7104 - 31)

= 7073 \times 5^6

Question 2:- How many ways 6 Boys & 6 Girls batch students can be seated around a circular table such that no two boys are together.

Solution:- See full solution

Question 3:- 2n chairs are arranged symmetrically around a table. there are 2n people including A and B, whose wish to occupy the chair. Find the number of seating arrangements if:-

(i) A and B are next to each

(ii) A and B are diametrical

Solution:- See full solution

Question 4:- Assuming cricket wordcup-2023 is played 10 teams. How many total matches are played in the league stage if each team plays with every other team exactly once

Solution:- See full solution

Question 5:- From a library containing 5 identical physics books, 4 identical chemistry books & 6 identical maths books. Find the number of ways in which we can select

(a) At least one book

(b) At least one book of each subject

(c) At least 2 books

(d) At least 2 book of each subject.

Solution:-See full solution

Question 6:-  In a hotel four rooms are available Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons . Then the number of all possible ways in which this can be done is ______

Solution:- See full solution

Exercise 7.3 ncert solutions maths class 11

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