Permutation & Combination concepts by Gaurav Sharma - Part (5/5)


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Find the total number of triangles with integral sides and perimeter 122

    Let A, B and C be the sides of a triangle.

    A + B + C = 122

    Also, A + B > C, B + C > A, A + C > B (sum of two sides > third side)

    So maximum value of any side will be (122/2) – 1 = 60

    S0, (60 – a) + (60 – b) + (60 – c) = 122  [0 ≤ a, b, c ≥ 59]

    a + b + c = 58

    So, total cases will be (58 + 3 – 1) C (3 – 1) = 60C2 = 30 x 59

    This will include cases where ( a = 19, b = 18, c = 21) , ( a = 18, b = 19, c = 21) … but all of the cases are forming the same triangles. So we need to subtract the repetitions.

    Now, a + b + c = 58

    1. Where any two of a, b and c are the same.

    2a + c = 58, here a can take values from 0 to 29 so 30 cases.

    Similarly for b = c and c = a hence total 30 x 3 = 90 cases.

    1. When a = b = c, not possible as a + b + c = 58

    So, cases where all are different will be (59 x 30) – 90 = 1680

    For distinct cases we will divide this by 3! = 6

    We will get 1680/6 = 280

    Cases where two are the same = 90

    For distinct cases we will divide this by 3, 90/3 = 30

    Total cases = 280 + 30 = 310

    Short cut:

    Total number of triangles with integral sides and perimeter n

    1. When n is odd : [ (n + 3)^2/48]
    2. Where n is even : [n^2/48]

    Where [.] is the nearest integer (Don’t confuse with greatest integer function)

    Applying for the above problem, [122^2]/48 = [310.08] = 310

    To find the number of scalene triangles with integral sides and perimeter n

    Put ( n – 6) instead of n in the above formula (for all triangles)

    1. When n is Odd : [(n – 3)^2/48]
    2. When n is even : [(n – 6) ^2/48]

    Where [.] is the nearest integer

    Example: Perimeter of a triangle is 150. Find the number of scalene triangles possible with integral sides

    Using short cut – n = 150 (even)

    Number of scalene triangles = [(150 – 6)^2/48] = 432

    Detailed approach:

    A + B + C = 150. Max value of any side can be 74

    So, (74 – a) + ( 74 – b) + (74 – c) = 150

    a + b + c = 72

    Total cases ( 72 + 3 – 1) C ( 3 – 1) = 74C2 = 37 x 73

    When two of them are same:

    2a + c = 72, where a goes from 0 to 36 but if a = 24 then b = c = a

    So 36 cases

    Similarly 36 cases for b = c and c = a each

    Also where a = b = c = 24 - > 1 case

    (37 x 73) – ( 36 x 3) – 1 = 2592

    Now divide this by 3! = 6 to find distinct cases

    Number of scalene triangles = 2592/6 = 432

    How many scalene triangles are possible if all the sides are integers and the perimeter of a triangle is 24 units?

    OA: 7

    Make sure that it is not greatest integer function here. We have to take nearest value

    The perimeter of a triangle is 150. Find the number of isosceles triangles possible with integral sides.

    A + B + C = 150

    Max value of any side can be 74

    So, (74 – a) + ( 74 – b) + (74 – c) = 150

    a + b + c = 72

    Total cases ( 72 + 3 – 1) C ( 3 – 1) = 74C2 = 37 x 73

    When two of them are same

    2a + c = 72, where a goes from 0 to 36 but if a = 24 then b = c = a

    So 36 cases

    Similarly 36 cases each for b = c and c = a each

    Total 36 x 3 = 108 triangles.

    But we have to exclude the repetitions

    So 108/3 = 36 triangles.

    Short cut:

    To find the number of isosceles triangles with integral sides and perimeter n

    No. of total triangles – no. of scalene triangles – no. of equilateral triangles

    Here perimeter = 150

    Total triangles = [n^2/48] = [468.75] = 469

    Scalene triangles = [(n – 6)^2/48] = [432] = 432

    Isosceles triangles = 469 – 432 – 1 = 36

    The number of integral even sided triangle with perimeter 180

    A + B + C = 180

    Max value of any side can be 89

    But sides are even so max value will be 88

    Also, A, B and C are even so put A = 88 – 2a, B = 88 – 2b, C = 88 – 2c

    (88 – 2a) + (88 – 2b) + (88 – 2c) = 180

    2(a + b + c) = 84

    a + b + c = 42

    Total cases : (42 + 3 – 1)C(3 – 1) = 44C2 = 946

    When two of them are same:

    2a + c = 42, where a goes from 0 to 21 but if a = 14 then a = b = c. so, 21 cases

    Similarly 21 cases for b = c and c = a each

    Also, where a = b = c = 24 – 1 case

    946 – ( 21 x 3) – 1 = 882

    Divide by 3! To get distinct cases

    Number of scalene triangles = 2592/6 = 147

    Number of isosceles triangles = 21 x 3 / 3 = 21

    Number of equilateral triangles = 1

    Total 147 + 21 + 1 = 169

    Short cut:

    Number of triangles with all sides (integral) EVEN

    Total number of triangles with integral sides and perimeter n (n is even) = n^2/48

    [When all sides are even perimeter will also be even]

    But, here we have to find number of triangles with all sides even so put n/2 instead of n in the above formula

    Required number of triangles – n^2/(4 x 48 )

    Here, perimeter = 180 and all sides should be even

    Number of triangles = [n^2/( 4 x 48 )] = [180^2/ ( 4 x 48 )] = [168.75] = 169

    Find the number of triangle having odd integral sides of perimeter equal to 153

    A+B+C=153
    max value can be 76
    put A = 76-(2a-1)
    similarly for B & C
    So , [ 76 - (2a-1) ]+ [ 76- (2b-1)]+[76-(2c-1)]= 153
    2(a+b+c)+3=76(3) - 153
    a+b+c= 36
    total (36+3-1)C(3-1) = 38C2 ways = 703 ways
    when a=b
    2a+c= 36
    a goes from 0 - 18 but this includes case where a = b = c
    so 19-1 = 18 cases
    similarly 18 cases for b = c & c = a each
    and 1 case of equilateral triangle

    Scalene - > (703-(38*3) - 1) /6 = 648/2 = 108
    isosceles - > 18
    equilateral - > 1
    total 108+18+1 = 127

    Or Apply formula (n + 3)^2/48 = [126.75] = 127

    The possible shortest routes in which we can travel from A to B (grid of m x n) are ( m + n )! / m!n!

    The possible shortest routest in which we can travel from A to B where there is a shortcut are:

    For example in grid of ( 6 x 4 )

    From A to C in 4!/2!2! = 6 ways

    From C to D = 1 way

    From D to B = 4!/1!3! = 4 ways

    Total ways = 6 x 1 x 4 = 24 ways

     



  • Find the number of triangle having odd integral sides of perimeter equal to 153

    what would be the direct formula for this @gaurav_sharma


  • Being MBAtious!


    @cat789

    Number of Triangles with Integer sides for a given perimeter.

    • If the perimeter p is even then, total triangles is [p^2]/48.
    • If the perimeter p is odd then, total triangles is [(p+3)^2]/48
    • If it asks for number of scalene triangle with a given perimeter P, then subtract 6 and apply the same formula . For even [(p-6)^2]/48 and for odd [(p-3)^2]/48.

    Where [x] represents neatest integer function. For example [6.7] is 7 not 6 because its nearest integer.


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