Question

How many different signals can be made by $5$ flags from $8$ flags of different colours?
A) $10$
B) $6720$
C) $20$
D) None of these

Answer 1

We are provided with $8$ flags of different colours and $5$ flags. We need to find the total number of signals that can be made by the number of arrangements of $8$ flags by taking $5$ flags at a time. Arranging things is called permutation, thus we can clearly say that the given question is based on the topic “permutation”. Here we have to arrange $8$ flags taken $5$ flags at a time which is equivalent to filling $8$ flags placed out of $5$ flags. For that we have the formula of permutation: $^n{P_r}$.

Complete step by step answer:
We have been told that there are $8$ flags of different colours by taking $5$ flags at a time. Thus the total number of flags $n = 8$ and flags to be taken at a time $r = 5$. By applying the permutation formula ${ = ^n}{P_r}$
${ = ^8}{P_5}$
By applying in permutation,
$= \dfrac{{8!}}{{(8 - 5)!}}$
Subtracting the values in denominator,
$= \dfrac{{8!}}{{3!}}$
Expanding the factorial,
$= \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1}}$
$3,2,1$ in both the numerator and the denominator will get cancel,
$= 8 \times 7 \times 6 \times 5 \times 4$
By multiplying all the terms we will get,
$= 6720$.
Therefore the required number of signals can be made by $5$ flags from $8$ flags of different colours is $6720$. Hence, option (B) is correct.

Note:
Each of the different arrangements which can be made by taking some or all of a number of things at a time is called permutation. The number of permutation without any repetition states that arranging $n$ objects taken $r$ at a time is equivalent to filling $r$ places out of $n$ things $= n(n - 1)(n - 2)...(n - \overline {r - 1} )$ ways $= \dfrac{{n!}}{{(n - r)!}}{ = ^n}{P_r}$. We can also say that, $^n{P_r} = n(n - 1)(n - 2)...(n - r + 1)$, thus, $^8{P_5} = 8 \times 7 \times 6 \times 5 \times 4 = 6720$.
Thus we can say that the permutation concerns both the selection and the arrangement of the selected things in all possible ways.