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

Hint: 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.