If a convex polygon has 44 diagonals, then find the number of its sides.
Let the convex polygon has $n$ sides.
$\therefore$ Number of diagonals $={ }^n C_2-n$
According to the question,
$$\begin{aligned} & { }^n C_2-n=44 \\ & \frac{n!}{2!(n-2)!}-n=44 \\ & \Rightarrow \quad \frac{n(n-1)}{2}-n=44 \\ & \Rightarrow \quad n\left[\frac{(n-1)}{2}-1\right]=44 \quad \Rightarrow n\left(\frac{n-1-2}{2}\right)=44 \\ & \Rightarrow \quad n(n-3)=44 \times 2 \Rightarrow n(n-3)=88 \\ & \Rightarrow \quad n^2-3 n-88=0 \quad \Rightarrow(n-11)(n+8)=0 \\ & \Rightarrow \quad n=11,-8 \\ & \therefore \quad n=11\quad [\because n \neq-8] \end{aligned}$$
18 mice were placed in two experimental groups and one control group with all groups equally large. In how many ways can the mice be placed into three groups?
It is given that 18 mice were placed equally in two experimental groups and one control group i.e., three groups.
$\therefore \quad$ Required arrangements $=\frac{\text { Total arrangement }}{\text { Equally likely arrangement }}=\frac{18!}{6!6!6!}$
A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag, if (i) they can be of any colour. (ii) two must be white and two red. (iii) they must all be of the same colour.
Total number of marbles $=6$ white +5 red $=11$ marbles
(i) If they can be of any colour means we have to select 4 marbles out of 11.
$\therefore$ Required number of ways $={ }^{11} C_4$
(ii) If two must be white, then selection will be ${ }^6 \mathrm{C}_2$ and two must be red, then selection will be ${ }^5 \mathrm{C}_2$.
$\therefore$ Required number of ways $={ }^6 C_2 \times{ }^5 C_2$
(iii) If they all must be of same colour, then selection of 4 white marbles out of $6={ }^6 C_4$
and selection of 4 red marble out of $5={ }^5 C_4$
$\therefore \quad$ Required number of ways $={ }^6 C_4+{ }^5 C_4$
In how many ways can a football team of 11 players be selected from 16 players? How many of them will
(i) include 2 particular players?
(ii) exclude 2 particular players?
Total number of players $=16$
We have to select a team of 11 players
(i) include 2 particular players $={ }^{16-2} C_{11-2}={ }^{14} C_9$
[since, selection of $n$ objects taken $r$ at a time in which pobjects are always included is ${ }^{n-p} C_{r-p}$ ]
(ii) Exclude 2 particular players $={ }^{16-2} C_{11}={ }^{14} C_{11}$
[since, selection of $n$ objects taken $r$ at a time in which p objects are never included is $\left.{ }^{n-p} C_r\right]$
A sports team of 11 students is to be constituted, choosing atleast 5 from class XI and atleast 5 from class XII. If there are 20 students in each of these classes, in how many ways can the team be constituted?
Total students in each class $=20$
We have to selects atleast 5 students from each class.
Hence, selection of sport team of 11 students from each class is given in following table
| Class XI | 5 | 6 |
|---|---|---|
| Class XII | 6 | 5 |
$\begin{aligned} \therefore \text { Total number of ways of selecting a team of } 11 \text { players } & ={ }^{20} C_5 \times{ }^{20} C_6+{ }^{20} C_6 \times{ }^{20} C_5 \\ & =2 \times{ }^{20} C_5 \times{ }^{20} C_6\end{aligned}$