Question

The least positive value of $x$ which satisfies the inequality $^{10}{C_{x - 1}} > {2^{10}}{C_x}$ is
(1) $7$
(2) $8$
(3) $9$
(4) $10$

Answer 1

Hint: A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter . i.e., we can select the items in any order . We know \[^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}\] is the formula of combination and we also know another one property of combination is $\dfrac{{^n{C_r}}}{{^n{C_{r - 1}}}} = \dfrac{{n - r + 1}}{r}$ . We use this property to solve the question.

Complete step by step answer:
From the given data $^{10}{C_{x - 1}} > {2^{10}}{C_x}$
To find the least positive value of integral value of $x$ which satisfy the inequality , we use property of combination
From the given inequality we get $^{10}{C_{x - 1}} > {2^{10}}{C_x}$
Dividing both sides of above inequality by $^{10}{C_{x - 1}}$ and get
$ \Rightarrow \dfrac{{{2^{10}}{C_x}}}{{^{10}{C_{x - 1}}}} < 1$
Use the property of combination $\dfrac{{^n{C_r}}}{{^n{C_{r - 1}}}} = \dfrac{{n - r + 1}}{r}$in the above inequality , we get
$ \Rightarrow 2 \times \dfrac{{10 - x + 1}}{x} < 1$
$ \Rightarrow 20 - 2x + 2 < x$
Calculate and we get
$ \Rightarrow 22 - 2x < x$
We subtract the function by $x$ both side , we get
$ \Rightarrow 22 - 3x < 0$
$ \Rightarrow 3x > 22$
Divide both side by $3$ , we get
$ \Rightarrow x > \dfrac{{22}}{3}$
$ \Rightarrow x = 7.333$
Therefore, the least positive value of $x$ is $7$ .
$\therefore $ Option (1) is correct.

Note:
We find the least positive integer means suppose we find the value in fraction and we take the least positive integer as a less than the fraction greatest integer . Suppose we get the fraction $9.34353$ , therefore the least positive integer is $9$ which is less than $9.34353$ .