...going out of the first room and into the second. 139. Formulas for Permutations. PROBLEM I. To find the number of permutations of n things taken all at a time. The number of permutations of n things taken all at a time is denoted by Pn. It is easy to see that...

...arrangements is then 6!/3! = 120. This example illustrates the following theorem. THEOREM. // P is the number of permutations of n things taken all at a time, of which ni are alike, n2 others alike, ns others alike, and so on, then n\ \ /is ! ^3 ! • • •...

...the number of permutations of n things taken r at a time is n(n -l)1n — 2)-.(n — r + 1). Сов. The number of permutations of n things taken all at a time is n(n — l)(n, — 2)— to n factors, or n(nl)(n-2)...3.2.1. It is usual to denote this product...

...that case, the n letters are all different, Pn" = \n . Therefore NX r X'PX 1 9 = i » . or \p That is, The number of permutations of n things, taken all at a time, when r of them are of one kind, p of another, I" and q of another, and so on, is -. — ^-rEXERCISE 140...

...1 things. Hence by the theorem of Section 131, nPr = n(n - 1) (n - 2) . . . (n - r + 1). Corollary. The number of permutations of n things taken all at a time is „/»„ = n(n - 1) (n - 2) . . . 2 x 1 = n! The symbol n! is read factional n, and represents...

...as obtained in § 8). Inasmuch as n\ is the result of placing r = n in formula (1), it follows that the number of permutations of n things taken all at a time is n\ Expressed as a formula, this result becomes (2) nPn=n(nl).-2-l = n! Thus, the five letters a,...

...n things are taken at once, the last factor in (87) becomes n — n + 1 which reduces to 1. Hence, the number of permutations of n things taken all at a time is nPn = n(n - l)(n - 2) ... 1 = n\ (88) The symbol n\ is read factorial n and denotes the product...

...factorial n and is denoted by the symbol n ! For example, 5 I means 5 • 4 • 3 • 2 • 1. Hence, The number of permutations of n things taken all at a time is factorial n. Exercises 1. In how many ways can the letters x, y, z be arranged ? Write each of these...

...product of all integers from n down to 1, which we have called factorial n (§ 53), we may say that the number of permutations of n things taken all at a time is given by the formula EXAMPLE 1. How many even numbers greater than 40,000 can be formed from the...

...second seat, any of the remaining four in the third seat, etc. There is a symbol used for describing "the number of permutations of n things taken all at a time." It is called the factorial, and we use the exclamation point to stand for it. Thus, n(n — I)(n —...