Digit arrangements forming numbers

5 Find how many different numbers can be made from some or all of the digits of the number 1345789 if

1. all seven digits are used, the odd digits are all together and no digits are repeated,
2. the numbers made are even numbers between 3000 and 5000, and no digits are repeated,
3. the numbers made are multiples of 5 which are less than 1000 , and digits can be repeated.

7 Nine cards are numbered \(1,2,2,3,3,4,6,6,6\).

1. All nine cards are placed in a line, making a 9-digit number. Find how many different 9-digit numbers can be made in this way
   (a) if the even digits are all together,
   (b) if the first and last digits are both odd.
2. Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how many different numbers can be made in this way
   (a) if there are no repeated digits,
   (b) if the number is between 200 and 300 .

6

1. 1. Find how many numbers there are between 100 and 999 in which all three digits are different.
   2. Find how many of the numbers in part (i) are odd numbers greater than 700 .
2. A bunch of flowers consists of a mixture of roses, tulips and daffodils. Tom orders a bunch of 7 flowers from a shop to give to a friend. There must be at least 2 of each type of flower. The shop has 6 roses, 5 tulips and 4 daffodils, all different from each other. Find the number of different bunches of flowers that are possible.

6

1. Find how many numbers between 3000 and 5000 can be formed from the digits \(1,2,3,4\) and 5,
   1. if digits are not repeated,
   2. if digits can be repeated and the number formed is odd.
2. A box of 20 biscuits contains 4 different chocolate biscuits, 2 different oatmeal biscuits and 14 different ginger biscuits. 6 biscuits are selected from the box at random.
   1. Find the number of different selections that include the 2 oatmeal biscuits.
   2. Find the probability that fewer than 3 chocolate biscuits are selected.

5

1. Find how many numbers between 5000 and 6000 can be formed from the digits 1, 2, 3, 4, 5 and 6
   1. if no digits are repeated,
   2. if repeated digits are allowed.
2. Find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls
   1. if there are more girls than boys in the team,
   2. if three of the boys are cousins and are either all in the team or all not in the team.

4

1. 1. Find how many different four-digit numbers can be made using only the digits 1, 3, 5 and 6 with no digit being repeated.
   2. Find how many different odd numbers greater than 500 can be made using some or all of the digits \(1,3,5\) and 6 with no digit being repeated.
2. Six cards numbered 1,2,3,4,5,6 are arranged randomly in a line. Find the probability that the cards numbered 4 and 5 are not next to each other.

3 Numbers are formed using some or all of the digits 4, 5, 6, 7 with no digit being used more than once.

1. Show that, using exactly 3 of the digits, there are 12 different odd numbers that can be formed.
2. Find how many odd numbers altogether can be formed.

6

1. Find the number of different 3-digit numbers greater than 300 that can be made from the digits \(1,2,3,4,6,8\) if
   1. no digit can be repeated,
   2. a digit can be repeated and the number made is even.
2. A team of 5 is chosen from 6 boys and 4 girls. Find the number of ways the team can be chosen if
   1. there are no restrictions,
   2. the team contains more boys than girls.

3 The digits 1, 2, 3, 4 and 5 are arranged in random order, to form a five-digit number.

1. How many different five-digit numbers can be formed?
2. Find the probability that the five-digit number is
   (a) odd,
   (b) less than 23000 .

6

1. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number.

   1

   3

   3

   3

   5

   5

   9

   How many different 7 -digit numbers can be formed using these cards?
2. The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it.
   \includegraphics[max width=\textwidth, alt={}, center]{98ac515d-fd47-4864-afd6-321e9848d6cb-04_398_801_596_632} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
   (a) How many selections of seven cards are possible?
   (b) Find the probability that the seven cards include exactly one card showing the letter A .

9 A bag contains 9 discs numbered 1, 2, 3, 4, 5, 6, 7, 8, 9 .

1. Andrea chooses 4 discs at random, without replacement, and places them in a row.
   (a) How many different 4 -digit numbers can be made?
   (b) How many different odd 4-digit numbers can be made?
2. Andrea's 4 discs are put back in the bag. Martin then chooses 4 discs at random, without replacement. Find the probability that
   (a) the 4 digits include at least 3 odd digits,
   (b) the 4 digits add up to 28 .

4

1. How many different 3-digit numbers can be formed using the digits 1, 2 and 3 when
   (a) no repetitions are allowed,
   (b) any repetitions are allowed,
   (c) each digit may be included at most twice?
2. How many different 4-digit numbers can be formed using the digits 1, 2 and 3 when each digit may be included at most twice?

3 Heidi has a pack of cards.
Each card has a single digit on one side and is blank on the other side.
Each of the digits from 1 to 9 appears on exactly four cards.
Apart from the numerical values of the digits, the cards are indistinguishable from each other.
Heidi draws four cards from the pack, at random and without replacement. She places the four cards in a row to make a four-digit number. Determine how many different four-digit numbers Heidi could have made in each of the following cases.

1. The four digits are all different.
2. Two of the digits are the same and the other two digits are different.
3. There is no restriction on whether any of the digits are the same or not.

1. 1. The polynomial \(\mathrm { F } ( x )\) is a quartic such that

$$\mathrm { F } ( x ) = p x ^ { 4 } + q x ^ { 3 } + 2 x ^ { 2 } + r x + s$$ where \(p , q , r\) and \(s\) are distinct constants.
Determine the number of possible quartics given that

1. the constants \(p , q , r\) and \(s\) belong to the set \(\{ - 4 , - 2,1,3,5 \}\)
2. the constants \(p , q , r\) and \(s\) belong to the set \(\{ - 4 , - 2,0,1,3,5 \}\)
   (ii) A 3-digit positive integer \(N = a b c\) has the following properties
   • \(N\) is divisible by 11
   • the sum of the digits of \(N\) is even
   • \(N \equiv 8 \bmod 9\)
   • Use the first two properties to show that
   $$a - b + c = 0$$
3. Hence determine all possible integers \(N\), showing all your working and reasoning.