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Chapter 1: Q.1.9 (page 15)

A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts the blocks in a line, how many arrangements are possible ?

Short Answer

Expert verified

Total possible arrangements are 27720

Step by step solution

01

.Given information

Here a child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. we have to find the possible arrangement if the child puts the blocks in a line

02

Step 2. finding the arrangements if the child put the blocks in a line 

If the child puts the 12 blocks in a line, then the different possible arrangements and from 12 blocks 6 are blacks means 6blocks are identical and 4 are red so 4 identical red blocks is the there and there is white and 1 blue block are there, .To get a different arrangement we have to divide 12! by identical arrangements .thus the arrangements will be 12!6!4!1!1!=12×11×10×9×8×7×6!4×3×2×1×6!=27720

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Most popular questions from this chapter

Suppose that 10fish are caught at a lake that contains 5distinct types of fish.

(a)How many different outcomes are possible, where an outcome specifies the numbers of caught fish of each of the 5types?

(b)How many outcomes are possible when3the 10fish caught are trout?

(c)How many when at least 2of the 10are trout?

In a certain community, there are 3 families consisting of a single parent and 1 child, 3 families consisting of a single parent and 2 children, 5 families consisting of 2 parents and a single child, 7 families consisting of 2 parents and 2 children, and 6 families consisting of 2 parents and 3 children. If a parent and child from the same family are to be chosen, how many possible choices are there?

Verify that the equality

∑x1+....+xr=n,xi≥0n!x1!x2!....xr!=rn

when n=3,r=2, and then show that it always valid. (The sum is over all vectors of r nonnegative integer values whose sum is n.)

Hint: How many different n letter sequences can be formed from the first r letters of the alphabet? How many of them use letter iof the alphabet a total of xitimes for each i=1,...,r?

Consider three classes, each consisting of n students. From this group of 3nstudents, a group of 3 students is to be chosen.

(a) How many choices are possible?

(b) How many choices are there in which all 3 students are in the same class?

(c) How many choices are there in which 2 of the 3 students are in the same class and the other student is in a different class?

(d) How many choices are there in which all 3 students are in different classes?

(e) Using the results of parts (a) through (d), write a combinatorial identity.

Consider a tournament of ncontestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let localid="1648231792067" N(n)denote the number of different possible outcomes. For instance, localid="1648231796484" N(2)=3, since, in a tournament with localid="1648231802600" 2contestants, player localid="1648231807229" 1could be uniquely first, player localid="1648231812796" 2could be uniquely first, or they could tie for first.

(a) List all the possible outcomes when n=3.

(b) With localid="1648231819245" N(0)defined to equal localid="1648231826690" 1, argue without any computations, that localid="1648281124813" N(n)=∑i=1nniNn-i

Hint: How many outcomes are there in which localid="1648231837145" iplayers tie for last place?

(c) Show that the formula of part (b) is equivalent to the following:

localid="1648285265701" N(n)=∑i=1n-1niNi

(d) Use the recursion to find N(3) and N(4).
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