Question

The number of ways in which four letters can be selected from the word ‘DEGREE’ is
a) 7
b) 6
c) \[\dfrac{{6!}}{{3!}}\]
d) None of these

Answer 1

Hint: Here in this question we have to select the four letters from the DEGREE and the letters can be arranged in different forms or patterns, there is no specific rule to arrange the letters. To select the letters we use the combination concept and using the formula of the combination we can determine the number of ways.

Complete answer:
A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, you can select the items in any order. The formula for the combination is given by \[{}^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}\]
On considering the given question, we have to select the four letters of the word DEGREE. In the word DEGREE there are 6 letters, where E is repeated thrice.
To select the letters we have 3 different ways.
i) 3 E’s are together and one letter is chosen from the other 3 letters and they are arranged in different positions.
Therefore the selection of 4 letters will be calculated by \[{}^3{C_1}\]
ii) 2 E’s are together and two letters are chosen from the other 3 letters and they are arranged in different positions.
Therefore the selection of 4 letters will be calculated by \[{}^3{C_2}\]
iii) All letters are arranged in different positions. Therefore the selection of 4 letters will be calculated by \[{}^3{C_3}\]
So the total number of total ways will be
\[ \Rightarrow {}^3{C_1} + {}^3{C_2} + {}^3{C_3}\]
On applying the formula we have
\[ \Rightarrow \dfrac{{3!}}{{2!1!}} + \dfrac{{3!}}{{1!2!}} + 1\]
On simplifying we have
\[ \Rightarrow 3 + 3 + 1\]
On adding we get
\[ \Rightarrow 7\,ways\]
Therefore in 7 ways we can select the 4 letters from the word DEGREE

Therefore, the correct option is a)

Note: Combinations can be confused with permutations. However, in permutations, the order of the selected items is essential. While in permutations, the arrangements are different. In combinations, you can select the items in any order. The formula for the \[n!\] is given by \[n! = n \times (n - 1) \times (n - 2) \times ... \times 3 \times 2 \times 1\]