Thea E. asked • 01/24/16
how many three digit numbers can be made with the digits 1,2,3,4,5,6 and 7 if a) repetitions are allowed b) no digits is to be repeated in a number? 1 Expert Answer With 7 numbers, we can create three digit numbers like this. a) With repetitions would suggest with replacement. We'd have 7 options for the first digit, 7 options for the 2nd digit and 7 choices for the 3rd digit b) Without repeated numbers, would suggest without replacement We'd have 7 options for the first digit, 6 options for the 2nd digit and 5 choices for the 3rd digit Text Solution Solution : (i) When repetition of digits is allowed: <br> No. of ways of choosing firsy digits = 5 <br> No. of ways of choosing second digit = 5 <br> No. of ways of choosing third digit = 5 <br> Therefore, total possible numbers `= 5 xx 5 xx 5 = 125` <br> (ii) When repetition of digits is not allowed: <br> No. of ways of choosing first digit = 5 <br> No. of ways of choosing second digit = 4 <br> No. of ways of choosing thrid digit = 3 <br> Total possible numbers `= 5 xx 4 xx 3 = 60`. Number System is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. It is a system of writing for expressing numbers. It gives a special representation to every number and represents the arithmetic and algebraic form of the number. It allows us to operate arithmetic operations like addition, subtraction, multiplication, and division. An equation is a statement that connects two algebraic expressions of the same values with the ‘=’ sign. For example: In equation 9x + 4 = 7, 9x + 4 is the left-hand side expression and 7 is the right-hand side expression connected with the ‘=’ sign. What is a Number? A word or symbol that indicates a quantity is known as a number. The numbers 2, 4, 6, etc. are even numbers and 1, 3, 5, etc. are odd numbers. A number is a value created by the merger of integers. These numbers are used to represent algebraic quantities. An integer is a sign from a set of 10 characters ranging from 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any combination of integers represents a number. The size of a Number depends on the count of digits that are used for its formation. For Example: 126, 128, 0.356, -12, 78, 94 etc. Numbers are of various types depending upon the patterns of digits that are used for their creation. Various symbols and rules are also applied to Numbers which classifies them into a variety of different types:
Permutations And Combinations Permutation is the different arrangements of a given number of components taken one by one, or some, or all at a time. For example, if we have two components X and Y, then there are two possible arrangements, XY and YX. The number of permutations when ‘r’ elements are arranged out of a total of ‘n’ elements is nPr = n!/(n – r)! For example, let n = 4 (A, B, C and D) and r = 2 (All permutations of size 2). The answer is 4!/(4-2)! = 12. So, the twelve permutations are AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, and DC. Combination is the different selections of a given number of components taken one by one, or some, or all at a time. For example, if we have two components A and B, then there is only one way to select two items, we select both of them. Number of combinations when ‘r’ elements are selected out of a total of ‘n’ elements is nCr = n!/[(r!) × (n – r)!] For example, let n = 4 (A, B, C and D) and r = 2 (All combinations of size 2). The answer is 4!/((4-2)! × 2!) = 6. So, the six combinations are AB, AC, AD, BC, BD, CD. nCr = nC(n –r)
(i) repetition of the digits is allowed? Solution:
(ii) repetition of the digits is not allowed? Solution:
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