PROBLEM IN THE DOCTRINE OF PERMUTATIONS. 2Q5 



rach of these last the sum of the q 4- l left terms, and so 

 ,on, through all the number of different things, and the laet 

 line will be the answer: that is, the second term shows the 

 number of combinations taking one at a time, the third term, 

 the number of combinations taking two at a time, &c 



Example, * 



Given a number of the form a* b 5 c* d 4 e*f* g> to find Example. 

 how many different divisors it lias, each of which shall be 

 the product of ten factors, of nine factors, of eight factors, 

 &c. ; a, 6, c, &e. being prime numbers. 



Here m z= 5, n zz 5, p zz 4, q ZZ 4, r ZZ 4, s zz 3, t zz 1, 

 therefore by the rules 



zz m -f- 1 units 

 \zzn +1 terms 

 15 — p + 1 terms 

 105—9 + 1 terms 

 15 15 35 70 123 193 275 3G0 435 486 ± r + 1 terms 

 1 6 21 56 125 243 421 66l 951 1263 1556 zzs f 1 terms 

 1 7 27 77 1B1 368 664 1082 l6l2 2214 2819 answers. 



That is, the number has seven prime divisions, twenty-seven 

 that are composed of two factors, seventy-seven having three 

 factors, &c. 



I have selected this question, because it includes the par- This rule com* 

 tkular case given by Emerson in his last example ; in order P ared wilh 

 that, by a comparison of both methods, an estimate may be 

 formed of the labour that is saved by this rule. It may not 

 at tie aame time be amiss to observe, that Emerson has not 

 put down a twentieth part of the work, that is necessary for 

 the operation. 



Investigation of the Rule. 



By the developement of the formula (l -f a -J- o • • • »a m ) Investigation ' 

 *( 1 + b + 6*. . . .//») .(Hc + c 2 -. -cP) • (l + d + d* of the rul<> 



• • • 'd?) &c, we shall evidently obtain all the possible com- 

 binations that can be formed with m a s, n b s, p cs, q ds, 

 &c. ; and, as we proceed in this developement, the law 

 whence the above ruU is deduced will be readily perceived. 



But, 



