207. In how many different ways can eight persons be arranged on eight seats?

Answer, 40320.

In how many ways can eight persons be seated at a round table, so that all shall not have the same neighbours in any two arrangements?[30]

Answer, 5040.

If the hundredth part of a farthing be given for every different arrangement which can be made of fifteen persons, to how much will the whole amount?

Answer, £13621608.

Out of seventeen consonants and five vowels, how many words can be made, having two consonants and one vowel in each?

Answer, 4080.

208. If two or more of the counters have the same letter upon them, the number of distinct permutations is less than that given by the last rule. Let there be a, a, a, b, c, d, and, for a moment, let us distinguish between the three as thus, a, a′, a″. Then, abca′a″d, and a″bcaa′d are reckoned as distinct permutations in the rule, whereas they would not have been so, had it not been for the accents. To compute the number of distinct permutations, let us make one with b, c, and d, leaving places for the as, thus, ( ) bc ( ) ( ) d. If the as had been distinguished as a, a′, a″, we might have made 3 × 2 × 1 distinct permutations, by filling up the vacant places in the above, all which six are the same when the as are not distinguished. Hence, to deduce the number of permutations of a, a, a, b, c, d, from that of aa′a″bcd, we must divide the latter by 3 × 2 × 1, or 6, which gives