Question

The number of different signals which can be given from \[7\] different coloured sheets, taking one or more at a time is-
A.$127$
B.$5913$
C.$13699$
D.$13700$

Answer 1

Hint: Here since we have to select the number of different signals given from $7$ different coloured sheets we will use the formula of combination. The number of ways to select r things from n things is given as-${}^n{C_r}$ And we also know that ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
Where n is the total number of things and r is the number of things to be selected. Use this formula to select one or more different coloured sheets to generate a signal. Add them all and solve to get the answer.

Complete step-by-step answer:
We have to find the number of different signals which can be given from $7$ different coloured sheets. We can take one coloured sheet or more than one coloured sheet.
So we know that the number of ways to select r things from n things is= ${}^n{C_r}$
And we also know that ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
Where n is the total number of things and r is the number of things to be selected.
Now on taking one coloured sheet at a time, the number of signals we can generate =${}^7{C_1}$
On taking two coloured sheets at a time, the number of signals we can generate=${}^7{C_2}$
On taking three coloured sheets at a time, the number of signals we can generate =${}^7{C_3}$
On taking four coloured sheets at a time, the number of signals we can generate= ${}^7{C_4}$
On taking five coloured sheets at a time, the number of signals we can generate= ${}^7{C_5}$
On taking six coloured sheets at a time, the number of signals we can generate= ${}^7{C_6}$
On taking seven coloured sheets at a time, the number of signals we can generate= ${}^7{C_7}$
So the totals number of ways we can generate different signals from seven different coloured sheets= ${}^7{C_1} + {}^7{C_2} + {}^7{C_3} + {}^7{C_4} + {}^7{C_5} + {}^7{C_6} + {}^7{C_7}$
Now on applying the formula of combination, we get-
The number of ways we can generate different signals from seven different coloured sheets=$\dfrac{{7!}}{{6!1!}} + \dfrac{{7!}}{{5!2!}} + \dfrac{{7!}}{{4!3!}} + \dfrac{{7!}}{{3!4!}} + \dfrac{{7!}}{{2!5!}} + \dfrac{{7!}}{{1!6!}} + \dfrac{{7!}}{{0!7!}}$
Now we know that $n! = n\left( {n - 1} \right)...3,2,1$
On applying this formula, we get-
The number of ways we can generate different signals from seven different coloured sheets= $\dfrac{{7 \times 6!}}{{6!}} + \dfrac{{7 \times 6 \times 5!}}{{5!2!}} + \dfrac{{7 \times 6 \times 5 \times 4!}}{{4!3 \times 2 \times 1}} + \dfrac{{7 \times 6 \times 5!}}{{2!5!}} + \dfrac{{7 \times 6!}}{{1!6!}} + \dfrac{1}{1}$
On simplifying we get,
The number of ways we can generate different signals from seven different coloured sheets=$7 + 21 + 35 + 21 + 7 + 1$
On addition we get,
The number of ways we can generate different signals from seven different coloured sheets=
$127$
Hence the correct answer is A.

Note: Here we are not using permutation because in this question we are not concerned with the order of the sheets selected. The combination is only concerned with selection not order while in permutation order is important. The permutation is concerned with the arrangement of things and it is given as-
${}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$
Where n is the total number of things and r is the number of things to be selected.