Question

The permutations and combinations of $abcd$ taken $3$ at a time are respectively.

Answer 1

Step 1. Calculate the permutations.

The number of permutations of $n$ objects taken $r$ at a time is given by,

$P(n,r)=\frac{n!}{(n-r)!}$

We have to find permutations of $abcd$ taken $3$ at a time.

$$\begin{gathered} \therefore P(4,3)=\frac{4!}{(4-3)!} \\ \Rightarrow =\frac{4\times 3\times 2\times 1}{1} \\ \Rightarrow =24 \end{gathered}$$

Step 2. Calculate the combination.

The number of combinations of $n$ objects taken $r$ at a time is given by,

$C(n,r)=\frac{n!}{r!(n-r)!}$

We have to find combinations of $abcd$ taken $3$ at a time.

$$\begin{gathered} \therefore C(4,3)=\frac{4!}{3!(4-3)!} \\ =\frac{4\times 3\times 2\times 1}{3\times 2\times 1\times 1} \\ =4 \end{gathered}$$

Hence, the permutation and combination are $24$ and $4$respectively.