What is the number of selections of at most 3 things from 6...

The number of ways to select $r$ things from $n$ different things is given by ^{n}C_{r}$.<br><br>The phrase "at most 3 things" means we can select 0, 1, 2, or 3 things from the 6 available different things.<br><br>The total number of selections is the sum of these possibilities:<br><br>^{6}C_{0} + ^{6}C_{1} + ^{6}C_{2} + ^{6}C_{3}$

Calculating each term:

^{6}C_{0} = 1$<br><br>^{6}C_{1} = 6$

^{6}C_{2} = $\dfrac{6 \times 5}{2 \times 1} =15$

^{6}C_{3} = $\dfrac{6 \times 5 \times 4}{3 \times 2 \times 1} =20$

Total number of selections = 1 + 6 + 15 + 20 = 42$.<br><br>Answer: 42$