Arrangement of Similar Items (TOMATO Subjective 49)

Arrangement of Similar Items (TOMATO Subjective 49)

Problem: \(x\) red balls, \(y\) black balls,\(z\) white balls are to be arranged in a row. Suppose that any two balls of the same color are indistinguishable. Given \(x+y+z=30\) prove that number of possible arrangements is maximum when \(x=y=z=10\). Solution: Given a...
The Gifts Distribution (TOMATO Subjective 48)

The Gifts Distribution (TOMATO Subjective 48)

Problem: Find the different number of ways \(5\) different gifts can be presented to \(3\) children so that each child receives at least one gift. Solution: There are two possible ways in which the gifts can be distributed. Case 1: They are distributed as \(2,2,1\)....
Dictionary Ranking (TOMATO Subjective 50)

Dictionary Ranking (TOMATO Subjective 50)

Problem: All the permutation of the letters \(a,b,c,d,e\) are written down and arranged in alphabetical order as in dictionary. Thus the arrangement \(abcde\) is in first position and \(abced\) is in second position. What is the position of the word \(debac\)?...

Number of Onto Functions (TOMATO Subjective 46)

Problem: A function \(f\) from set \(A\) into set \(B\) is a rule which assigns each element \(x\) in \(A\), a unique (one and only one) element (denoted by \(f(x)\) in \(B\). A function of set from \(A\) into \(B\) is called an onto function, if for each element...
Symmetrical Minima. (TOMATO Subjective 33)

Symmetrical Minima. (TOMATO Subjective 33)

Problem: Let \(k\) be a fixed odd positive integer. Find the minimum value of \(x^2 + y^2\), where \(x,y\) are non-negative integers and \(x+y=k\).   Solution: We have \(y=k-x\). Therefore we get an equation in \(x\) where \(k\) is a constant, precisely \(f(x) = x^2 +...