Question

How many words with or without meaning of three distinct letters of the English alphabet are there?

Answer 1

Hint: We will use the concept of finding possibilities by using a simple process. This is what we have applied here is called the concept of forming choices by considering an individual case first and then use it to move further to find the total cases.

Complete step-by-step answer:
Now we will first start by knowing the number of letters in an English alphabet. There are a total of 26 letters in English alphabets. According to the question we are supposed to form a three letters word out of these 26 letters. Also this is done without repetition as we are clearly given that the letters should be distinct while forming a three letter word. So this is done as by selecting any three letters say, A, B, J out of 26 letters and words formed may be ABJ, JBA, AJB, JAB, BJA, BAJ. As we cannot do this for so many letters now, we need to form a three letter word. For that we have 26 choices for being the first letter of the word. As in this question repetition is not allowed therefore for choosing a second letter we are left with 25 choices. And for the last one we have 24 choices. As forming the word was a continued process. Therefore, we will multiply all these choices together. Thus, we now have $26\times 25\times 24=15600$.
Hence, the total number of ways of forming the required words is 15600.

Note: Alternately we can solve it by using the permutation concept. In this concept we actually use the formula given as $P_{r}^{n}=\dfrac{n!}{\left( n-r \right)!}$ where n is the total number of objects and r is the selected objects out of n. In this question we have n = 26 and r = 3. Therefore we get
$\begin{align}
  & P_{3}^{26}=\dfrac{26!}{\left( 26-3 \right)!} \\
 & \Rightarrow P_{3}^{26}=\dfrac{26!}{\left( 23 \right)!} \\
 & \Rightarrow P_{3}^{26}=\dfrac{26\times 25\times 24\times 23!}{\left( 23 \right)!} \\
 & \Rightarrow P_{3}^{26}=\dfrac{26\times 25\times 24\times 1}{1} \\
 & \Rightarrow P_{3}^{26}=15600 \\
\end{align}$
Hence, the total number of ways of forming the required words is 15600.