GRE Quantitative: Data Analysis – Counting Methods

There are many methods through which students can get a better understanding of a data set out of which, Mean and Median are the simplest techniques of all. However when it comes to GRE, the exam tests students through different mathematical problems and these problems need more sophisticated methods; such as counting methods for solving when compared to the simple conventional techniques generally used. This article presents several counting methods that could be utilised for the GRE syllabus.

  • The Principle of Multiplication

The concept which is at the roots of the principle of multiplication can be considered to be the simplest counting method that you can witness in the GRE General Test. The Principle of Multiplication can be stated as follows:

Suppose there are l choices for one article and k options for another item, then the likely unique possibilities for the pair of articles will be l ×k or lk.

The Principle of Multiplication is applicable only if the choices of one article with respect to the other are entirely independent. This principle will also apply if there are many alternatives of many articles, as long as they are independent of one another.

For example : If Monica has 8 shirts, 5 skirts, 7 jeans, 3 jackets and 9 pairs of shoes then the total number of outfits she has equals to 8 × 5 × 7 × 3 × 9 = 7560

The Multiplication Principle can also be applied to the probabilistic situations where the possible outcomes are always two at each trial, and the trials are independent of one another. For example, if a coin is being tossed five times. In this case, there are only two possible outcomes (either heads or tails) in each trial.

Hence, total number of possible outcomes = 2 × 2 × 2 × 2 × 2 = 25 = 32

  • Factorials:

For a set of Natural numbers {1, 2, 3, 4, 5, …}, the factorial of n can be represented as n!, where n is a natural number, and it is defined as the product of all the natural numbers that are less than and equal to n.

For example; 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Factorials are mainly used for permutation and combination.

  • Permutation & Combination

Permutation: An ordered arrangement of objects is known as a permutation. The formula used for calculating permutations for r objects taken from n different objects is:
\(\large ^{n}P_{r} = \frac{n!}{(n -r)!}\)

Combination: Ways of selecting items from a group, in which the order of selection does not matter. The formula used for calculating combination is
\(\large ^{n}C_{r} = \frac{^{n}P_{r}}{r!} = \frac{n!}{r!(n -r)!}\)

Where n is the total number of choices and r is the select arrangements.

In questions where arrangements matter, use permutation, and on others use combination.

Question: A football team of 11 players have to be chosen from a total of 16 players. How many ways can you do that?

Solution: Here the total number is 16. So, n = 16.

We have to ‘select’ a team of 11 players. So r = 11.

Since we’ve to select, we should use the formula for Combination. Substitute the values in the formula to get the answers.

Question: A football team of 11 players has to be chosen from a total of 16 players. The first 5 players you choose should be the penalty takers of the team. How many ways can you do that?

Solution:Here the total number of players is 16, So n = 16.

We have to select a team of 11 players, but the first five will be the penalty takers.

Since the first five have to be in order, from the 11, we use the formula for Permutation. Substitute the values in the formula to obtain the answer.

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