You’re visiting a city with 7 major tourist attractions, and you only have time to visit 3 of them in one day. You want to plan the order in which you’ll visit them to make the most of your experience. You plan to spend equal time at each site, so the order doesn’t matter. How many options do you have for choosing 3 attractions?
If there are 7 attractions and we choose 3 of them. From permutations, there are \({}_7C_3 = 210\) different arrangements.
Since the order doesn’t matter, we divide 210 by the number of rearrangements, 3! = 6.
So, there are 210/6 = 35 different combinations.
Looking at the full calculation:
\[\begin{align*} {}_7C_3 &= \frac{7!}{3!(7-3)!} \\ &= \frac{7!}{3!4!} \\ &= \frac{~~~7\times 6\times 5\times 4\times 3\times 2\times 1}{(3\times 2\times 1)\times(4\times 3\times 2\times 1)} \\ &= \frac{7\times 6\times 5}{3\times 2\times 1} \\ &= \frac{210}{6}\\ &= \mathbf{35} \end{align*}\]