| a | , | b | , | c | , and | d | , |
| p | q | r | s |
that is, is less than the greatest of them, and greater than the least. Let these fractions be arranged in order of magnitude; that is, let a/p be greater than b/q, b/q be greater than c/r, and c/r greater than d/s. Then by (200)
| is less than | and greater than | |||||||
| a + b | a | b | and | c | ||||
| p + q | p | q | r | |||||
| a + b + c | a + b | and | a | c | and | d | ||
| p + q + r | p + q | p | r | s | ||||
| a + b + c + d | a + b + c | and | a | d | ||||
| p + q + r + s | p + q + r | p | s | |||||
whence the proposition is evident.
203. It is usual to signify “a is greater than b” by a > b and “a is less than b” by a < b; the opening of V being turned towards the greater quantity. The pupil is recommended to make himself familiar with these signs.
SECTION IX.
ON PERMUTATIONS AND
COMBINATIONS.
204. If a number of counters, distinguished by different letters, be placed on the table, and any number of them, say four, be taken away, the question is, to determine in how many different ways this can be done. Each way of doing it gives what is called a combination of four, but which might with more propriety be called a selection of four. Two combinations or selections are called different, which differ in any way whatever; thus, abcd and abce are different, d being in one and e in the other, the remaining parts being the same. Let there be six counters, a, b, c, d, e, and f; the combinations of three which can be made out of them are twenty in number, as follow:
| abc | ace | bcd | bef |
| abd | acf | bce | cde |
| abe | ade | bcf | cdf |
| abf | adf | bde | cef |
| acd | aef | bdf | def |
The combinations of four are fifteen in number, namely,
| abcd | abde | acde | adef | bcef |
| abce | abdf | acdf | bcde | bdcf |
| abcf | abef | acef | bcdf | cdef |