Answer
Verified
479.1k+ views
Hint: Calculate the number of ways to choose \[1\] officers from \[4\] officers. Calculate the number of ways to choose \[6-1=5\] jawans from \[8\] jawans. Multiply both the values to calculate the total number of ways to choose \[6\] people.
Complete step-by-step answer:
We have a group of \[4\] officers and \[8\] jawans. We have to choose \[6\] people such that it includes exactly one officer. We have to find the number of ways to do so.
We will calculate the number of ways to choose one officer from \[4\] officers. Then we will calculate the number of ways to choose remaining people from \[8\] jawans. We will then multiply both the values to calculate the total number of ways to choose \[6\] people.
We know there are \[{}^{n}{{C}_{r}}\] ways to choose \[r\] people from a group of \[n\] people. We know that \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\].
Substituting \[n=4,r=1\], we have \[{}^{4}{{C}_{1}}=\dfrac{4!}{1!\left( 3 \right)!}=\dfrac{4\times 3!}{3!}=4\] ways to choose one officer from \[4\] officers.
As we have to choose total \[6\] people and we have already chosen one officer, the number of jawans to be chosen \[=6-1=5\]. So, we will now choose \[5\] jawans from \[8\] jawans.
Substituting \[n=8,r=5\], we have \[{}^{8}{{C}_{5}}=\dfrac{8!}{5!\left( 3 \right)!}=\dfrac{8\times 7\times 6\times 5!}{5!\times 3!}=\dfrac{8\times 7\times 6}{3\times 2}=56\] ways to choose \[5\] jawans from \[8\] jawans.
To calculate the total number of ways to choose \[6\] people according to the given data, we will multiply both the values of choosing one officer from \[4\] officers and \[5\] jawans from \[8\] jawans.
Thus, the total number of ways to choose \[6\] people according to the given data \[=4\times 56=224\].
Hence, we can choose \[6\] people such that it includes exactly one officer is \[224\].
Note: In this question, we are basically calculating all the possible combinations to choose people. One must keep in mind that we are not considering the arrangement of chosen people on this question. If we count the arrangement of people, we will get an incorrect answer.
Complete step-by-step answer:
We have a group of \[4\] officers and \[8\] jawans. We have to choose \[6\] people such that it includes exactly one officer. We have to find the number of ways to do so.
We will calculate the number of ways to choose one officer from \[4\] officers. Then we will calculate the number of ways to choose remaining people from \[8\] jawans. We will then multiply both the values to calculate the total number of ways to choose \[6\] people.
We know there are \[{}^{n}{{C}_{r}}\] ways to choose \[r\] people from a group of \[n\] people. We know that \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\].
Substituting \[n=4,r=1\], we have \[{}^{4}{{C}_{1}}=\dfrac{4!}{1!\left( 3 \right)!}=\dfrac{4\times 3!}{3!}=4\] ways to choose one officer from \[4\] officers.
As we have to choose total \[6\] people and we have already chosen one officer, the number of jawans to be chosen \[=6-1=5\]. So, we will now choose \[5\] jawans from \[8\] jawans.
Substituting \[n=8,r=5\], we have \[{}^{8}{{C}_{5}}=\dfrac{8!}{5!\left( 3 \right)!}=\dfrac{8\times 7\times 6\times 5!}{5!\times 3!}=\dfrac{8\times 7\times 6}{3\times 2}=56\] ways to choose \[5\] jawans from \[8\] jawans.
To calculate the total number of ways to choose \[6\] people according to the given data, we will multiply both the values of choosing one officer from \[4\] officers and \[5\] jawans from \[8\] jawans.
Thus, the total number of ways to choose \[6\] people according to the given data \[=4\times 56=224\].
Hence, we can choose \[6\] people such that it includes exactly one officer is \[224\].
Note: In this question, we are basically calculating all the possible combinations to choose people. One must keep in mind that we are not considering the arrangement of chosen people on this question. If we count the arrangement of people, we will get an incorrect answer.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE