There are 4 balls of different colours and 4 boxes of colours same as those of the balls, The number of ways in which the balls, one in each box could be placed such that a ball does not go to a box of its own colour is (a) 5 (b) 6 (c) 9 (d) 12

When we talk about permutations and combinations, we are referring to the various ways objects can be arranged or selected. Permutation is the arrangement of objects in a specific order, while combination is the selection of objects without regard for order.

To illustrate, suppose you have three books: A, B, and C. If you want to arrange these books on a shelf (and order matters), there are six permutations: ABC, ACB, BAC, BCA, CAB, and CBA. On the other hand, if you're only interested in selecting two books to take on a trip and order doesn't matter, there are three combinations: AB, AC, and BC.

These concepts are widespread in probability and statistics, as they are fundamental in calculating the likelihood of various outcomes. The key to solving problems involving permutation and combination is to determine whether order is an important factor and then apply the appropriate formulas.

The concept of a factorial is pivotal in permutations and combinations, as well as in derangement problems. A factorial, denoted by an exclamation point (!), is the product of all positive integers up to a given number. For instance, the factorial of 4 (written as 4!) is calculated as:
\( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Factorials grow exponentially with the size of the number, which is why they are critical when determining the total number of permutations. In the scenario given in the exercise, where we are dealing with 4 different elements, we use factorials to determine the number of ways to order or arrange these elements under specific constraints.

Reordering sets with constraints, like in our derangement problem, introduces an extra layer of complexity to permutations. A derangement is a type of permutation where none of the elements appear in their original positions. This is a common puzzle in combinatorics, a branch of mathematics dealing with the counting, arrangement, and combination of objects.

Moreover, when calculating derangements, a factorial is modified by a series of fractions with alternating signs, representing the inclusion-exclusion principle, to exclude permutations where any element is in its original place. As the exact number of conditions not to be met increases, the calculation becomes more complex and interesting. This principle is crucial when solving problems like the one given, where the balls must not end up in the boxes of the same color, forcing us to calculate specific permutations with constraints.