<p>Can someone explain this SAT Math Question. The wording really confuses me and I'm not really sure what it is asking.
[This can be found on pg 519 of the CollegeBoard 2nd Edition SAT Study Guide]</p>
<p>18) Any 2 points determine a line. If there are 6 points in a plane, no 3 of which lie on the same line, how many lines are determined by pairs of these 6 points?</p>
<p>It is 6C2= 30. Because you are finding out how many ways you can select exactly two points from six. It is simple because you don’t have to worry about colinear points</p>
<p>The answer is 15, not 30. And you didnt explain your answer.
Can someone give me a thorough explanation?</p>
<p>If no three points are collinear, then that basically means that you’re finding how many combinations of 2 points are possible, since every unique pair of points is part of a unique line. So to do this, you can use the combinations formula 6C2. The number of ways to choosing r of n objects is n!/((n-r)!r!). Plugging 6 and 2 into this formula, you get:</p>
<p>6!/(4!2!)</p>
<p>(6*5)/2! (just cancelled out 4! in numerator and denominator)</p>
<p>30/2 = 15. (simplify)</p>
<p>I hope this wasn’t overly confusing…</p>
<p>Elf, that is the formula for permutation not combination. Combination is n!/(n-r)!</p>
<p>chess you’re wrong.</p>
<p>@chess123, you want combination. Order of which you select the two points is irrelevant.</p>
<p>6C2 = 6!/(4!*2!) = 15 ← this is a combination.</p>
<p>A permutation assumes the order matters.</p>
<p>chess is wrong, he has it opp.</p>
<p>just write it out
1,2; 1,3; 1,4; 1,5; 1,6;
2,3; 2,4; 2,5; 2,6;
3,4; 3,5; 3,6;
4,5; 4,6;
5,6
= 15</p>
<p>its a permutation without repetition so it is increasing by one option each time as shown above. 6! 5! 4! 3! 2! 1!</p>
<p>If you are not comfortable with combinations, and happen to have more of a “geometry” mind, you can solve this quickly by visualizing any hexagon. </p>
<p>While it is good to know/memorize the number of diagonals in a geometric form, you could also remember the formula. </p>
<p>Either way knowing that a hexagon has NINE diagonals is helpful. Then all you need to do is add the six lines that form the hexagon. </p>
<p>9 + 6 = 15. </p>
<p>PS The explanation is more complex than “seeing it” and knowing the diagonal formula that is ½n(n–3). Play with it a bit, and this will be a nice addition to your SAT toolbox.</p>
<p>Or you could think of it this way: Pick a point out of the 6. There are 5 points you can draw a line to. Now pick another point. There are four new lines, one less since the line through your new point and original point is already drawn. Continuing this way, we get 5+4+3+2+1=15</p>
<p>decreasing not increasing</p>
<p>all that math is cool, it really is, but just drawing all dots and counting lines would be easier :)</p>
<p>^Yeah, unless you have 20 points.</p>
<p>that would be a problem beyond the scope of the SAT test</p>