Chapter 1: Q. 1.20 (page 19)

In how many ways can $n$ identical balls be distributed into $r$ urns so that the $i t h$ urn contains at least $m_{i}$ balls, for each $i = 1, . . ., r$ ? Assume that $n \geq \sum_{i = 1}^{r} m_{i}$ .

Short Answer

Expert verified

The required number of ways are = $n - \sum_{r - 1} m + r - 1$ .

Step by step solution

Step 1. Given information.

It is given that,

No. of identical balls = $n$

No. of urns = $r$

The $i t h$ urn contains at least $m_{i}$ balls, where $i = 1, 2, 3, . . . . . ., r$ .

Step 2. Find the required no. of ways.

Firstly, we distribute $m_{i}$ balls in the $i t h$ urn, where $i = 1, 2, 3, . . . . . ., r$ .

So, the no. of balls that have been distributed $= \sum_{i = 1}^{r} m_{i}$

The remaining balls arerole="math" localid="1648301319689" $= n - \sum_{i = 1}^{r} m_{i}$

By proposition 6.2, there are $(\begin{matrix} n + r - 1 \\ r - 1 \end{matrix})$

The distinct non-negative integer valued vectors $(x_{1}, x_{2}, . . . . ., x_{r})$ satisfying $x_{1} + x_{2} + . . . . . + x_{r} = n$

Similarly in this case we have to distribute $n - \sum_{i = 1}^{r} m_{i}$ balls in role="math" localid="1648301285511" $r$ urns.

Therefore, it can be done in $n - \sum_{r - 1} m + r - 1$ ways.

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In how many ways can $n$ identical balls be distributed into $r$ urns so that the $i t h$ urn contains at least $m_{i}$ balls, for each $i = 1, . . ., r$ ? Assume that $n \geq \sum_{i = 1}^{r} m_{i}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the required no. of ways.

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In how many ways can nidentical balls be distributed into rurns so that the ithurn contains at least miballs, for each i=1,...,r? Assume that n≥∑i=1rmi.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the required no. of ways.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Statistics

Pure Maths

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Study anywhere. Anytime. Across all devices.

In how many ways can $n$ identical balls be distributed into $r$ urns so that the $i t h$ urn contains at least $m_{i}$ balls, for each $i = 1, . . ., r$ ? Assume that $n \geq \sum_{i = 1}^{r} m_{i}$ .