Consider three boxes, each containing 10 balls labelled 1,2, …, 10. Suppose one ball is randomly

Name: Problem 7, Chapter 7: 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics
Uploaded: 2021-11-04T11:51:27-08:00
Duration: 3 min 54 s
Channel: Sriparna Bhattacharjee
Description: Consider three boxes, each containing 10 balls labelled 1,2, …, 10. Suppose one ball is randomly drawn from each of the boxes. Denote by ni, the label of the ball drawn from the i^th box, (i=1,2,3). Then, the number of ways in which the balls can be chosen such that n1<n2 <n3 is: (a) 120 (b) 82 (c) 240 (d) 164

step 1 In the question they are saying, to consider three boxes, each containing 10 balls, label 1 to 1

Question

Consider three boxes, each containing 10 balls labelled $1,2, \ldots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $\mathrm{n}_{i}$, the label of the ball drawn from the $i^{\text {th }}$ box, $(i=1,2,3)$. Then, the number of ways in which the balls can be chosen such that $\mathrm{n}_{1}<\mathrm{n}_{2}$ $<\mathrm{n}_{3}$ is: (a) 120 (b) 82 (c) 240 (d) 164

   Consider three boxes, each containing 10 balls labelled $1,2, \ldots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $\mathrm{n}_{i}$, the label of the ball drawn from the $i^{\text {th }}$ box, $(i=1,2,3)$. Then, the number of ways in which the balls can be chosen such that $\mathrm{n}_{1}<\mathrm{n}_{2}$ $<\mathrm{n}_{3}$ is:
(a) 120
(b) 82
(c) 240
(d) 164

43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics

Disha Experts 16th Edition

Chapter 7, Problem 7

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Solved by Expert Sriparna Bhattacharjee

11/04/2021

Step 1

We are to draw one ball from each box such that the label of the ball drawn from the first box is less than the label of the ball drawn from the second box, which in turn is less than the label of the ball drawn from the third box. Show more…

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Consider three boxes, each containing 10 balls labelled $1,2, \ldots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $\mathrm{n}_{i}$, the label of the ball drawn from the $i^{\text {th }}$ box, $(i=1,2,3)$. Then, the number of ways in which the balls can be chosen such that $\mathrm{n}_{1}<\mathrm{n}_{2}$ $<\mathrm{n}_{3}$ is: (a) 120 (b) 82 (c) 240 (d) 164

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Key Concepts

Combinations

This concept involves selecting a subset of items from a larger set where the order of selection does not matter. In combinatorial problems, the number of ways to choose k items from a set of n items is given by the binomial coefficient, commonly notated as C(n, k) or n choose k.

Binomial Coefficient

This is a mathematical way to count the number of combinations. It is defined as C(n, k) = n! / (k!(n-k)!), where n! denotes the factorial of n. This coefficient plays a pivotal role in problems that require counting selections from a set without regard to order.

Ordering Constraints

In many combinatorial problems, restrictions on the order of elements, such as requiring that selected numbers be in increasing order, simplify counting. An ordering constraint like n1 < n2 < n3 means that once the items are chosen, their arrangement is fixed, effectively reducing the problem to a combination selection rather than a permutation problem.

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