Question

What is the probability that six letters randomly selected from the alphabet will be picked in alphabetical order? A. \(\frac{3}{13}\) B. \(\frac{1}{20}\) C. \(\frac{1}{120}\) D. \(\frac{1}{720}\)

Short answer

The probability that six letters randomly selected from the alphabet will be picked in alphabetical order is D. \( \frac{1}{720} \).

Step-by-step solution

- Understand the problem

We need to find the probability that six letters randomly selected from the alphabet will be in alphabetical order.

- Calculate the total number of possible selections

There are 26 letters in the alphabet. The total number of ways to choose 6 letters from 26, without regard to order, is given by the combination formula: \(\binom{26}{6} = \frac{26!}{6!(26-6)!}\) Calculate this value.

- Calculate the number of favorable outcomes

A favorable outcome occurs when the 6 chosen letters are in alphabetical order. There is exactly one way to arrange any 6 letters out of 26 in alphabetical order.

- Find the probability

The probability of a favorable outcome is the ratio of the number of favorable outcomes to the total number of outcomes. So, the probability is: \(\frac{1}{\binom{26}{6}}\) Substitute \( \binom{26}{6} \) from the previous step and calculate the result.

- Simplify the fraction

Substituting \( \binom{26}{6} = 230230 \), the probability is: \(\frac{1}{230230} \) This value is approximately \( \frac{1}{720} \), which corresponds to option D.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations of objects belonging to a finite set. When solving problems involving combinations, we often use the binomial coefficient formula. This is expressed as \(\binom{n}{k}\), which represents the number of ways to choose \(k\) elements from a total of \(n\) elements without regard to the order of selection.

For instance, to find out how many different groups of 6 letters we can pick from the 26 letters of the alphabet, we use the formula: \[ \binom{26}{6} = \frac{26!}{6!(26-6)!} \] This value calculates the possible ways to select 6 letters from 26, considering the order doesn't matter.

Understanding how to apply combinatorial principles is essential to solve probability problems effectively.

Alphabetical Order

Alphabetical order means arranging items in the same sequence as they appear in the alphabet. When dealing with probabilities of sequences, identifying whether a set needs to be in alphabetical order can simplify the calculations.

In our problem, once we select any 6 letters from the alphabet, there is exactly one way to arrange them such that they follow the alphabetical order.

For example, if you pick the letters A, D, F, Q, W, and Z, the only alphabetical arrangement is A, D, F, Q, W, Z. Consequently, each unique combination of 6 letters has exactly one 'favorable' ordering when it comes to arranging them alphabetically.

Favorable Outcomes

In probability theory, a favorable outcome refers to the desired result from all possible outcomes. To determine the probability of these favorable outcomes, we calculate the ratio of favorable outcomes to the total number of outcomes.

In our exercise, the favorable outcome is having the 6 selected letters in alphabetical order. Only one specific arrangement of any chosen 6 letters will be in alphabetical order. Thus, there is 1 favorable outcome per each combination of 6 letters chosen from the 26-letter alphabet.

The total number of possible ways to choose 6 letters is \(\binom{26}{6} = 230230\). Hence, the probability of randomly picking any 6 letters and having them in alphabetical order is:

\[ P = \frac{1}{230230} \]

This simplifies to approximately \( \frac{1}{720} \), indicating how highly unlikely the event is.