In how many different ways can a chemistry book, a calculus book, a history book and a dictionary be arranged on a shelf so that the chemistry book or the history book appears first?
you have 4 books
One of two specific books have to come first, after that it, it doesn't matter, so
number of ways = 2 x 3 x 2 x 1 = 12
To find the number of different ways the books can be arranged on the shelf with the condition that the chemistry book or the history book appears first, we can use the concept of permutations.
First, let's consider the chemistry book or the history book as a single entity (let's call it "CH"). Now we have three entities to arrange on the shelf: CH, the calculus book, and the dictionary.
The number of ways to arrange these three entities is 3!, which is the factorial of 3 and equals 3 × 2 × 1 = 6. This accounts for all the possible arrangements of just the calculus book, the dictionary, and the CH entity.
However, we must also consider the two possible positions for the CH entity in relation to the other books (chemistry book first or history book first). So, we need to multiply the number of arrangements by 2, resulting in a total of 6 × 2 = 12 different ways.
Therefore, there are 12 different ways the chemistry book, calculus book, history book, and dictionary can be arranged on the shelf so that the chemistry book or the history book appears first.