Question

The total combinations of picking $3$ balloons from a packet of $25$ balloons are ________

Answer 1

Find the combination:

We have to find the combinations of picking $3$ balloons from a packet of $25$ balloons.
Therefore we find the combination ${}^{25}{\mathrm{C}}_3$
We use the formula of the combination i.e., ${}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}=\frac{\mathrm{n}!}{\mathrm{r}!(\mathrm{n}-\mathrm{r})!}$ , we get
${}^{23}{\mathrm{C}}_3=\frac{25!}{3!(25-3)!}$
Expanding the above factorial and we get
$={\displaystyle \frac{25\times 24\times 23\times 22!}{3\times 2\times 1\times 22!}}$
Now we omitting the same term and we get
$={\displaystyle \frac{25\times 24\times 23}{3\times 2}}$
Simplifying the denominator and we get
$={\displaystyle \frac{25\times 24\times 23}{6}}$
Now we divide $24$ by $6$ , therefore we get ${\displaystyle \frac{24}{6}}=4$
Put this in the above function and we get
$=25\times 23\times 4$
Multiplying $25$ and $4$ , we get
$=100\times 23$
Simplifying and we get
$=2300$
Hence,the total combinations of picking $3$ balloons from a packet of $25$ balloons are $2300$.