Question: The number of arrangements of the letters a b c d in which neither a, b nor c, d come together. A)...

Question

The number of arrangements of the letters a b c d in which neither a, b nor c, d come together.
A) 6
B) 12
C) 16
D) None of these

Answer 1

Solution

In this question first we find the total number of ways to arrange 4 letters after that we find the number of possible ways in which ${\text{a & b}}$ and ${\text{c & d}}$ together. After that, we subtract it from the total number of ways and we will get the required answer.

Complete step by step solution: As we all know letters a b c d can be arranged in $4!$
$4! = 4 \times 3 \times 2 \times 1$
$\Rightarrow 4! = 24$
$\Rightarrow$ Total possible arrangement of letters a b c d is 24.
In order to calculate neither ${\text{a & b}}$ nor ${\text{c & d}}$ come together first we calculate the cases when ${\text{a & b}}$ comes together and ${\text{c & d}}$ together.
So, take ${\text{a & b}}$ as one entity so we have total 3 letters.
$\Rightarrow$ 3 letters can be arranged in $3!$ and ${\text{a & b}}$ itself arranged in $2!$
$\Rightarrow$ Case when ${\text{a & b}}$ comes together then letters can be arranged in $3! \times 2!$$$$ = 3 \times 2 \times 1 \times 2 \times 1 = 12$ ways.
Similarly,
Take ${\text{c & d}}$ as one entity so we have total 3 letters.
$\Rightarrow$ 3 letters can be arranged in $3!$ and ${\text{c & d}}$ itself arranged in $2!$
$\Rightarrow$ Case when ${\text{c & d}}$ comes together then letters can be arranged in $3! \times 2!$$$$ = 3 \times 2 \times 1 \times 2 \times 1 = 12$ ways.
Now, the case when ${\text{a & b}}$ and ${\text{c & d}}$ both come together.
$\Rightarrow$ Case when ${\text{a & b}}$ and ${\text{c & d}}$ both come together then letters can be arranged in $2! \times 2! \times 2! = 2 \times 2 \times 2 = 8$ ways.
So, the total number of arrangement when neither ${\text{a & b}}$ nor ${\text{c & d}}$ come together $= 24 - 12 - 12 + 8$
$= 8$
8 number of ways the letters a b c d can arrange in which neither a,b nor c,d come together.

Hence, option D. None of these is the correct answer.

Note: Here we have used the concept of permutation. Permutation can be defined as arranging or rearranging the elements of a set. It gives the number of ways a certain number of objects can be arranged. Combinations can be defined as selecting a number of elements from a given set. Both permutations and combinations are similar except that in permutations, the order is important and in combinations the order is not important.