The total number of 5-digit numbers, then, is 4 в™ґ! = 4 в™І4 = 96.Д¬) How many 5-digit odd numbers can you make with 0, 1, 2, 3, 4, andДЄgain, 0 cannot be first, so remove it. Therefore, there will be 4 ways to fill the second spot, 3 ways to fill the third, and so on. It will now be one of 4 remaining digits. Then there will be 4 ways to choose the first digit. a) How many permutations are there of the digits 01234?Д«) How many 5-digit numbers can you make of those digits, in whichД«) the first digit is not 0, and no digit is repeated?

Consequently, the total number of arrangements in which n, u, and m are together, is 3! But in each one of them, there are 3! rearrangements of num. Therefore, there are 6! such arrangements.Д¬) How many have b as the last letter-or in any specified position?Д) How many will have n, u, and m together?Д«egin by permuting the 5 things- num, b, e, r, s. Set b as the first letter, and permute the remaining 6. a) How many different arrangements are there of the letters of the word numbers?Д«) How many of those arrangements have b as the first letter? It is the number of permutations of 5 different things taken 3 at a time. How many different arrangements (permutations) are there of the digits 34567?ДЄ) If the five letters a, b, c, d, e are put into a hat, in how many differentД«) After one of them has been drawn our, in how many ways could youД¬) Therefore, in how many ways could you draw two letters?Д) What is the meaning of the symbol 5 P 3? How many permutations are there of the letters pqrs? To cover the answer again, click 'Refresh' ('Reload'). To see the answer, pass your mouse over the colored area. The definition 0! = 1 makes line ( 1) above valid for all values of k: kВ =В 0,В 1,В 2. N P n is the number of permutations of n different things taken n at a time-it is the total number of permutations of n things: n!. This is the number of permutations of 10 different things taken 4 at a time.Д®xample 4. When the 6!'s cancel, the fraction reduces to 10 в™№ в™ё в™·. The upper factorial is the upper index, and the lower factorial is the difference of the indices. The upper factorial is that of the upper index of P, while the lower factorial is the difference of the indices.Д®xample 3. We see, then, that 8 P 3 can be expressed in terms of factorials as follows: The upper index 4 indicates the first factor.ДЇor example, 8 P 3 means: 'The number of permutations of 8 different things taken 3 at a time.' 8 P 3ДЇor, there are 8 ways to choose the first, 7 ways to choose the second, and 6 ways to choose the third.ДҐ! is a factor of 8!, and therefore the 5!'s cancel. The lower index 2 indicates the number of factors. 'The number of permutations of 4 different things taken 2 at a time.' We have seen that the number of ways of choosing 2 letters Therefore, the total number of ways they can be next to each other is 2 Then we will be permuting the 5 things qe, s, u a, r. There are 5! such permutations.Д«) Let q and e be next to each other as qe. After that has happened, there are 4 ways to fill the third, 3 to fill the fourth, and so on. Then there are 5 ways to fill the first spot.

There are 6! permutations of the 6 letters of the word square.ДЄ) In how many of them is r the second letter? _ r _ _ _ _Д«) In how many of them are q and e next to each other?ДЄ) Let r be the second letter. In how many different ways could you arrange them?Д®xample 2. The number of permutations of n different things taken n at a timeД®xample 1. We mean, '4! is the number of permutations of all 4 of 4 different things.') (To say 'taken 4 at a time' is a convention.

Thus the number of permutations of 4 different things taken 4 at a time is 4!. Therefore the number of permutations of 4 different things is 3 ways remain to choose the second, 2 ways to choose the third, and 1 way to choose the last. Let us now consider the total number of permutations of all four letters.

abДЄb means that a was chosen first and b second ba means that b was chosen first and a second and so on.

3 or 12 possible ways to choose two letters from four.

That is, to each of those possible 4 there will correspond 3. After that has happened, there will be 3 ways to choose the second. We can draw the first in 4 different ways: either a or b or c or d. If something can be chosen, or can happen, or be done, in m different ways, and, after that has happened, something else can be chosen in n different ways, then the number of ways of choosing both of them is m ДЇor example, imagine putting the letters a, b, c, d into a hat, and then drawing two of them in succession.

26 permute 3