Permutation and Combination: Advanced

Permutation & Combination form an important part of Quant section in CAT and other MBA entrance exams. In this article, we will discuss the advanced concepts of Permutation & Combination and formulae required for solving problems on the same. As the questions from this topic appear in all the major competitive exams, so you cannot skip this topic. This article will equip you to correctly solve complex permutation and combination questions, appearing in examinations.

Combination formula

• If the number of non negative integral solutions for the equation x₁ + x₂ + x₃ + x₄....x_r = n is^{n + r - 1} C _{r - 1}. In this case, value of any variable can be zero.
• If the number of positive integral solutions for the equation x₁ + x₂ + x₃ + x₄....x_r = n is ^{n - 1} C _{r - 1}. In this case, minimum value for any variable is 1.

Permutation Formula

• The number of non negative integral solutions for the equation x₁ + x₂ + x₃ + x₄....x_r = n, given the variable cannot have equal values and minimum value for any variable is 1, is given by,^{n - 1} P _{r - 1}

Solved Permutation and Combination Problems:

Example 1: In how many ways can 10 similar balls be put in 4 distinct boxes?

Solution: This question can be understood as finding the number of non-negative integral solutions to the equation w + x + y + z = 10. Using the above mentioned formula, we can get the answer as^{10 + 4 - 1}C_{4 - 1} = ¹³C₃. Solving this, we get the answer as 286.

Example 2: In how many ways can at most 10 similar balls be put in 4 distinct boxes?

Solution: This question can be understood as finding the number of non-negative integral solutions to the equation a + b + c + d = 10. So, let us introduce another variable 'e' in the equation to make the total as 10. Now, the equation becomes a + b + c + d + e = 10. Using the above mentioned formula, we can get the answer as^{10 + 5 - 1}C_{5 - 1} = ¹⁴C₄. Solving this, we get the answer as 1001. This formula works as the fifth variable taken i.e. 'e' can take any value, so make the sum of all the five items as 10.

Example 3: In how many ways can 10 similar balls be put in 4 distinct boxes such that each box contains at least 1 ball?

Solution: This question is almost similar to the 1^st question with a minor difference. In this question, each box must have at least 1 ball. This question can also be understood as finding the number of positive integral solutions to the equation w + x + y + z = 10, where minimum values of w, x, y and z is 1. So we will give value of 1 to each of these 4 variables. So new equation becomes w + x + y + z = 6. Using the formula given above, we get the answer as ^{6 + 4 - 1} C _{4 - 1} = ⁹C₃= 84. Or we can use the direct formula ^{n - 1} C _{r - 1}. Putting n = 10 and r = 4, we get the answer as ⁹C₃= 84.

Permutation and Combination: Key Learning

• The prime learning of this article is to find the number of non-negative integral solutions using the concept of Permutation and Combination. While these questions are rephrased using "atmost", "atleast" etc. but the basic formula on the reframed equation remains the same.