is and less greater _a_ + _b_ _a_ than than _b_ _c_ --------- --- --- and --- _p_ + _q_ _p_ _q_ _r_ _a_ + _b_ + _c_ _a_ + _b_ _a_ _c_ _d_ --------------- --------- and --- --- and --- _p_ + _q_ + _r_ _p_ + _q_ _p_ _r_ _s_ _a_ + _b_ + _c_ + _d_ _a_ + _b_ + _c_ _a_ _d_ --------------------- --------------- and --- --- _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_ and so on. 205. Each of these combinations may be written in several different orders; thus, _abcd_ may be disposed in any of the following ways: _abcd_ _acbd_ _acdb_ _abdc_ _adbc_ _adcb_ _bacd_ _cabd_ _cadb_ _badc_ _dabc_ _dacb_ _bcad_ _cbad_ _cdab_ _bdac_ _dbac_ _dcab_ _bcda_ _cbda_ _cdba_ _bdca_ _dbca_ _dcba_ of which no two are entirely in the same order. Each of these is said to be a distinct _permutation_ of _abcd_. Considered as a _combination_, they are all the same, as each contains _a_, _b_, _c_, and _d_.