Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
121 views
in Permutations by (24.8k points)
closed by

There are 10 lamps in a hall. Each one of them can be switched on independently. find the number of ways in which the hall can be illuminated.

1 Answer

+1 vote
by (27.0k points)
selected by
 
Best answer

We have to find the possible number of ways in which we can illuminate the hall by lighting the lamps which are independent of each other. 

We will use the concept of multiplication because there are four sub jobs dependent on each other and are performed one after the other. 

We will use a combination technique in this question, which will make us solve the question in an easy way. The combination means selection in rough language. The combination is denoted by the symbol νCr, which means a number of ways of selecting r objects at a time from n objects. 

We will make cases by choosing a specific number of lamps at one time,

Taking one lamp at a time, 

We can enlighten the hall in 10C1 ways. 

Taking two lamps at a time, 

We can enlighten the hall in 10C2 ways. 

Taking three lamps at a time, 

We can enlighten the hall in 10C3 ways 

Taking four lamps at a time, 

We can enlighten the hall in 10C4 ways. 

Taking five lamps at a time, 

We can enlighten the hall in 10C5 ways. 

Taking six lamps at a time, 

We can enlighten the hall in 10C6 ways. 

Taking seven lamps at a time,

We can enlighten the hall in 10C7 ways. 

Taking eight lamps at a time, 

We can enlighten the hall in 10C8 ways. 

Taking nine lamps at a time,

We can enlighten the hall in 10C9 ways. 

Taking ten lamps at a time, 

We can enlighten the hall in 10C10 ways. 

By using the binomial expansion (1+x)10,

If we put x = 1, then 

We will get the sum of these binomial coefficients in which there is extra 10C0 term which is equal to 1, so we subtract 1 from the sum. 

Therefore,

The total number of ways of enlightening the hall are = 210 – 1

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

...