Answer
Verified
477.6k+ views
Hint: - Here we choose three sides from n sides of a polygon by method of selection to form the triangle i.e.${}^n{C_3}$.Then similarly do for the n+1 sides of the polygon. After that apply the condition of the question.
Complete step by step solution:
Given that,
${T_{n + 1}} - {T_n} = 21$
${T_{n + 1}}$ Can be written as ${}^{n + 1}{C_3}$
$ \Rightarrow {}^{(n + 1)}{C_3} - {}^n{C_3} = 21$
$ \Rightarrow {}^n{C_2} + {}^n{C_3} - {}^n{C_3} = 21$
$\because $ We know that ${}^{(n + 1)}{C_r} = {}^n{C_{r - 1}} + {}^n{C_r}$
$ \Rightarrow \dfrac{{n!}}{{2!\left( {n - 2} \right)!}} = 21$
$\because $We know that ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
$ \Rightarrow \dfrac{{n \times (n - 1) \times (n - 2)!}}{{2 \times (n - 2)!}} = 21$
$
\Rightarrow {n^2} - n = 42 \\
\Rightarrow {n^2} - n - 42 = 0 \\
\Rightarrow {n^2} - 7n + 6n - 42 = 0 \\
\Rightarrow n(n - 7) + 6(n - 7) = 0 \\
\Rightarrow (n - 7)(n + 6) = 0 \\
$
$\therefore $ n=7 0r n=-6
We know that sides cannot be negative $\therefore n = 7$ is the required answer.
Hence, option B is the correct answer.
Note:-Whenever we face such a type of question the key concepts for solving the question is that you have to first choose the three sides from the n sides by selection method to form the triangle and then proceed according to the condition which is given in the question.
Complete step by step solution:
Given that,
${T_{n + 1}} - {T_n} = 21$
${T_{n + 1}}$ Can be written as ${}^{n + 1}{C_3}$
$ \Rightarrow {}^{(n + 1)}{C_3} - {}^n{C_3} = 21$
$ \Rightarrow {}^n{C_2} + {}^n{C_3} - {}^n{C_3} = 21$
$\because $ We know that ${}^{(n + 1)}{C_r} = {}^n{C_{r - 1}} + {}^n{C_r}$
$ \Rightarrow \dfrac{{n!}}{{2!\left( {n - 2} \right)!}} = 21$
$\because $We know that ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
$ \Rightarrow \dfrac{{n \times (n - 1) \times (n - 2)!}}{{2 \times (n - 2)!}} = 21$
$
\Rightarrow {n^2} - n = 42 \\
\Rightarrow {n^2} - n - 42 = 0 \\
\Rightarrow {n^2} - 7n + 6n - 42 = 0 \\
\Rightarrow n(n - 7) + 6(n - 7) = 0 \\
\Rightarrow (n - 7)(n + 6) = 0 \\
$
$\therefore $ n=7 0r n=-6
We know that sides cannot be negative $\therefore n = 7$ is the required answer.
Hence, option B is the correct answer.
Note:-Whenever we face such a type of question the key concepts for solving the question is that you have to first choose the three sides from the n sides by selection method to form the triangle and then proceed according to the condition which is given in the question.
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