Inclusion-Exclusion Principle and Applications
... Proof of Theorem 6.3.1 Let S be the set of all permutations of {1, 2, . . . , n} : |S| = n! For 1 ≤ j ≤ n let A j be the permutations i1 i2 . . . in with i j = j. Then Dn = |A1 ∩ A2 ∩ · · · ∩ An |. What are the permutations in A1 ∩ A2 ∩ · · · ∩ Ak ? They are of the form i1 i2 . . . ik ik+1 . . . in ...
... Proof of Theorem 6.3.1 Let S be the set of all permutations of {1, 2, . . . , n} : |S| = n! For 1 ≤ j ≤ n let A j be the permutations i1 i2 . . . in with i j = j. Then Dn = |A1 ∩ A2 ∩ · · · ∩ An |. What are the permutations in A1 ∩ A2 ∩ · · · ∩ Ak ? They are of the form i1 i2 . . . ik ik+1 . . . in ...
Document
... digit to the left of the decimal point (less then 10 but greater than 1) The second factor is a power of ten x ...
... digit to the left of the decimal point (less then 10 but greater than 1) The second factor is a power of ten x ...
Whole Numbers & Integers
... When you add a positive and negative number, it is like you are actually subtracting. For example (+7) + (-3) = (+4). The number line shows how to add a positive and a negative ...
... When you add a positive and negative number, it is like you are actually subtracting. For example (+7) + (-3) = (+4). The number line shows how to add a positive and a negative ...
form 2- 24 roots and radicles - kcpe-kcse
... If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator. This process involves multiplying the quotient by a form of 1 that will eliminate the radical in the ...
... If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator. This process involves multiplying the quotient by a form of 1 that will eliminate the radical in the ...
Rules For Significant Figures
... What are Significant Figures? The significant figures in a measurement Consist of all the digits known with certainty plus one final digit, which is uncertain or is estimated. ...
... What are Significant Figures? The significant figures in a measurement Consist of all the digits known with certainty plus one final digit, which is uncertain or is estimated. ...
Question 4: (marks)
... 8 × 9 × 10 × 11 × 12 × 13 × 14 is equal to another such product of consecutive whole numbers. What is the smallest and largest numbers of this other product? ______________________________________________________________________________ Question 12: [2 marks] A quadrilateral ABCD has AB = AD = 5, CB ...
... 8 × 9 × 10 × 11 × 12 × 13 × 14 is equal to another such product of consecutive whole numbers. What is the smallest and largest numbers of this other product? ______________________________________________________________________________ Question 12: [2 marks] A quadrilateral ABCD has AB = AD = 5, CB ...
solving Inequalities by Multiplying or Dividing
... Translation: 3times x is less than or equal to 12. ...
... Translation: 3times x is less than or equal to 12. ...
A New Connection Between the Triangles of Stirling and Pascal
... number! That’s true, Pascal’s triangle doesn’t always yield perfect numbers in this manner, but every even perfect number does appear somewhere in this sequence. This is because the number of elements in each white triangle is given by 2n –1(2n – 1). With n = 1, we get 1. Making n = 2 or 3 gives 6 a ...
... number! That’s true, Pascal’s triangle doesn’t always yield perfect numbers in this manner, but every even perfect number does appear somewhere in this sequence. This is because the number of elements in each white triangle is given by 2n –1(2n – 1). With n = 1, we get 1. Making n = 2 or 3 gives 6 a ...