Question

How many two letter words can be formed using letters from the word SPACE, when repetition of letters (i) is allowed, (ii) is not allowed?

Answer 1

Hint: The way is when you want 2 letters you have 5 choices for each step. When repetition is allowed both places have 5 choices. When repetition is not allowed one will have 5 choices, the other will have 4 choices. Now look at the definition of combination. Find the number of ways for each letter. Now use product rules to calculate the total number of ways.

Combination: It is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of selection does not matter in combination you can select items in any order. Formula is given by $^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$

Complete step-by-step answer:
Given condition in question can be written in form of:
2 letter word SPACE (i) with repetition (ii) no repetition

Case-I Solving the question with repetition.
The number of way possible to select first letter is given as:
$^{5}{{C}_{1}}$
By simplifying this above one, we get value of way as: 5
The number of ways possible for second letter are given by (as repetition is allowed the letters selected before is also available)
$^{5}{{C}_{1}}$
By simplifying this above one, we get value of ways are: 5

Rule of product: In combinations the rule of product or multiplication principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. Given P, Q works can be done at a time. Total number of ways to do both P, Q works are given by $\left( A\times B \right)$ ways.
By using product rule, we can get the total number of ways $5\times 5$
By simplifying, we can say that the number of ways is:
Number of ways with repetition= 25.

Case- II Solving the question without repetition.
The number of ways possible to select first letter are:
$^{5}{{C}_{1}}$
By simplifying the above term, we get value of ways as 5
The number of ways possible to select second letter are:
(As repetition is not there first letter selected is not available – 4 choices)
$^{4}{{C}_{1}}$
By simplifying the above term, we get the total ways as 4.

Rule of product: In combinations, the rule of product or multiplication principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. Given P, Q works can be done at a time. Total number of ways to do both P, Q work are given by $\left( A\times B \right)$ ways.
By using the product rule, we can get the total number of ways:
$\left( 5\times 4 \right)$
By simplifying, we can say that the number of ways are:
20
Therefore, with and without repetition we get 25 and 20 ways respectively.

Note: Don’t confuse between 2 cases. Without repetition means no second chance for the same letter. Students reverse these cases and solve. So, be careful with cases. Alternative way is finding without repetition and then add the cases of repetition (SS, PP, AA, CC, EE). So, if we add “5” we get our result for the repetition case.