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Binomial Coefficients &
Pascal’s TriangleDr Aamir Hashmi
Email: aamir.shahzad@ucp.edu.pk
Announcements• Assignment-4 will be issued tomorrow
• Quiz -3 (about Assignment-3) will be open on Friday
• Class-Activity-4, on Thursday ….
Binomial Coefficients &
Pascal’s TriangleDr Aamir Hashmi
Email: aamir.shahzad@ucp.edu.pk
Recap
Binomial Coefficient Identities
When we choose k objects out of n, in how many ways we can choose?𝑛𝑘=
𝑛!
𝑘! 𝑛 − 𝑘 !
Recap
Recall Combinations
When we choose k objects out of n, in how many ways we can choose?𝑛𝑘=
𝑛!
𝑘! 𝑛 − 𝑘 !
Most importantly, these combinations are symmetric:𝑛𝑘=
𝑛𝑛 − 𝑘
Recap
Recall Combinations
When we choose k objects out of n, in how many ways we can choose?𝑛𝑘=
𝑛!
𝑘! 𝑛 − 𝑘 !
Most importantly, these combinations are symmetric:
𝑛𝑘=
𝑛𝑛 − 𝑘
103
=107
=10!
3! 7!= 120
Recap
Recall Combinations
When we choose k objects out of n, in how many ways we can choose?𝑛𝑘=
𝑛!
𝑘! 𝑛 − 𝑘 !
Most importantly, these combinations are symmetric:
𝑛𝑘=
𝑛𝑛 − 𝑘
𝑛𝑘=
𝑛!
𝑘! 𝑛−𝑘 !and
𝑛𝑛 − 𝑘
=𝑛!
𝑛−𝑘 ! 𝑘!
Binomial Theorem𝑥 + 𝑦 𝑛
Binomial Theorem
or
𝒙 + 𝒚 𝒏 = 𝒌=𝟎𝒏 𝒏
𝒌𝒙𝒏−𝒌𝒚𝒌
Binomial Theorem(Some familiar expressions)
𝒙 + 𝒚 𝟓 = 𝒙 + 𝒚 ⋅ 𝒙 + 𝒚 ⋅ 𝒙 + 𝒚 ⋅ 𝒙 + 𝒚 ⋅ 𝒙 + 𝒚
𝒙 + 𝒚 𝟓 =𝟓𝟎𝒙𝟓 +
𝟓𝟏𝒙𝟒𝒚 +
𝟓𝟐𝒙𝟑𝒚𝟐 +
𝟓𝟑𝒙𝟐𝒚𝟑 +
𝟓𝟒𝒙𝒚𝟒 +
𝟓𝟓𝒚𝟓
𝒙 + 𝒚 𝟓 = 𝒙𝟓 + 𝟓 𝒙𝟒𝒚 + 𝟏𝟎 𝒙𝟑𝒚𝟐 + 𝟏𝟎 𝒙𝟐𝒚𝟑 + 𝟓 𝒙𝒚𝟒 + 𝒚𝟓
1st Factor 2nd Factor 3rd Factor 4th Factor 5th Factor
Pascal’s Triangle
𝒙 + 𝒚 𝟎 = 𝟏
𝒙 + 𝒚 𝟏 = 𝒙 + 𝒚
𝒙 + 𝒚 𝟐 = 𝒙𝟐 + 𝟐𝒙𝒚 + 𝒚𝟐
𝒙 + 𝒚 𝟑
𝒙 + 𝒚 𝟒
𝒙 + 𝒚 𝟓
𝒙 + 𝒚 𝟔
𝒙 + 𝒚 𝟕
𝒙 + 𝒚 𝟖
𝒙 + 𝒚 𝟗
𝒙 + 𝒚 𝟏𝟎
Pascal’s Triangle
Pascal’s TriangleThe 𝑛𝑡ℎ row contains the numbers
𝑛0,𝑛1,𝑛2, ⋯ ,
𝑛𝑛
𝑛 = 0, 0th row
𝑛 = 1, 1st row
𝑛 = 2, 2nd row
𝑛 = 3, 3rd row
𝑛 = 4, 4th row
𝑛 = 5, 5th row
𝑛 = 6, 6th row
Pascal’s Triangle
Pascal’s TriangleThe core of this TRIANGLE is the Pascal’s Identity:
Every number in it is the sum of the two numbers immediately above ite.g.
Pascal’s TriangleThe core of this TRIANGLE is the Pascal’s Identity:
Every number in it is the sum of the two numbers immediately above ite.g.
Pascal’s TriangleThe core of this TRIANGLE is the Pascal’s Identity:
Proof of Pascal’s identity:LHS: 𝑛
𝑘=
𝑛!
𝑘! 𝑛−𝑘 !
RHS: 𝑛 − 1𝑘 − 1
+𝑛 − 1𝑘
=𝑛−1 !
𝑘−1 ! 𝑛−1−𝑘+1 !+
𝑛−1 !
𝑘 ! 𝑛−1−𝑘 !
Pascal’s TriangleThe core of this TRIANGLE is the Pascal’s Identity:
Proof of Pascal’s identity:LHS: 𝑛
𝑘=
𝑛!
𝑘! 𝑛−𝑘 !
RHS: 𝑛 − 1𝑘 − 1
+𝑛 − 1𝑘
=𝑛−1 !
𝑘−1 ! 𝑛−1−𝑘+1 !+
𝑛−1 !
𝑘 ! 𝑛−1−𝑘 !
= 𝑛 − 1 !1
𝑘 − 1 ! 𝑛 − 𝑘 !+
1
𝑘 ! 𝑛 − 𝑘 − 1 !
Pascal’s TriangleThe core of this TRIANGLE is the Pascal’s Identity:
Proof of Pascal’s identity:LHS: 𝑛
𝑘=
𝑛!
𝑘! 𝑛−𝑘 !
RHS: 𝑛 − 1𝑘 − 1
+𝑛 − 1𝑘
=𝑛−1 !
𝑘−1 ! 𝑛−1−𝑘+1 !+
𝑛−1 !
𝑘 ! 𝑛−1−𝑘 !
= 𝑛 − 1 !1
𝑘 − 1 ! 𝑛 − 𝑘 !+
1
𝑘 ! 𝑛 − 𝑘 − 1 !
= 𝑛 − 1 !1
𝑘 − 1 ! (𝑛 − 𝑘) 𝑛 − 𝑘 − 1 !+
1
𝑘 𝑘 − 1 ! 𝑛 − 1 − 𝑘 !
Pascal’s TriangleThe core of this TRIANGLE is the Pascal’s Identity:
Proof of Pascal’s identity:LHS: 𝑛
𝑘=
𝑛!
𝑘! 𝑛−𝑘 !
RHS: 𝑛 − 1𝑘 − 1
+𝑛 − 1𝑘
=𝑛−1 !
𝑘−1 ! 𝑛−1−𝑘+1 !+
𝑛−1 !
𝑘 ! 𝑛−1−𝑘 !
= 𝑛 − 1 !1
𝑘 − 1 ! 𝑛 − 𝑘 !+
1
𝑘 ! 𝑛 − 𝑘 − 1 !
= 𝑛 − 1 !1
𝑘 − 1 ! (𝑛 − 𝑘) 𝑛 − 𝑘 − 1 !+
1
𝑘 𝑘 − 1 ! 𝑛 − 1 − 𝑘 !
=𝑛 − 1 !
𝑘 − 1 ! 𝑛 − 𝑘 − 1 !
1
(𝑛 − 𝑘)+1
𝑘
Pascal’s TriangleThe core of this TRIANGLE is the Pascal’s Identity:
Proof of Pascal’s identity:LHS: 𝑛
𝑘=
𝑛!
𝑘! 𝑛−𝑘 !
RHS: 𝑛 − 1𝑘 − 1
+𝑛 − 1𝑘
=𝑛−1 !
𝑘−1 ! 𝑛−1−𝑘+1 !+
𝑛−1 !
𝑘 ! 𝑛−1−𝑘 !
= 𝑛 − 1 !1
𝑘 − 1 ! 𝑛 − 𝑘 !+
1
𝑘 ! 𝑛 − 𝑘 − 1 !
= 𝑛 − 1 !1
𝑘 − 1 ! (𝑛 − 𝑘) 𝑛 − 𝑘 − 1 !+
1
𝑘 𝑘 − 1 ! 𝑛 − 1 − 𝑘 !
=𝑛 − 1 !
𝑘 − 1 ! 𝑛 − 𝑘 − 1 !
1
(𝑛 − 𝑘)+1
𝑘
=𝑛 − 1 !
𝑘 − 1 ! 𝑛 − 𝑘 − 1 !
𝑘 + 𝑛 − 𝑘
𝑘(𝑛 − 𝑘)=
𝑛 − 1 !
𝑘 − 1 ! 𝑛 − 𝑘 − 1 !
𝑛
𝑘(𝑛 − 𝑘)
Pascal’s TriangleThe core of this TRIANGLE is the Pascal’s Identity:
Proof of Pascal’s identity:LHS: 𝑛
𝑘=
𝑛!
𝑘! 𝑛−𝑘 !
RHS: 𝑛 − 1𝑘 − 1
+𝑛 − 1𝑘
=𝑛−1 !
𝑘−1 ! 𝑛−1−𝑘+1 !+
𝑛−1 !
𝑘 ! 𝑛−1−𝑘 !
= 𝑛 − 1 !1
𝑘 − 1 ! 𝑛 − 𝑘 !+
1
𝑘 ! 𝑛 − 𝑘 − 1 !
= 𝑛 − 1 !1
𝑘 − 1 ! (𝑛 − 𝑘) 𝑛 − 𝑘 − 1 !+
1
𝑘 𝑘 − 1 ! 𝑛 − 1 − 𝑘 !
=𝑛 − 1 !
𝑘 − 1 ! 𝑛 − 𝑘 − 1 !
1
(𝑛 − 𝑘)+1
𝑘
=𝑛 − 1 !
𝑘 − 1 ! 𝑛 − 𝑘 − 1 !
𝑘 + 𝑛 − 𝑘
𝑘(𝑛 − 𝑘)=
𝑛 − 1 !
𝑘 − 1 ! 𝑛 − 𝑘 − 1 !
𝑛
𝑘(𝑛 − 𝑘)
=𝒏!
𝒌! 𝒏−𝒌 != LHS
Pascal’s TriangleThe core of this TRIANGLE I the Pascal’s Identity:
Every number in it is the sum of the two numbers immediately above ite.g.
Pascal’s TriangleThe core of this TRIANGLE I the Pascal’s Identity:
Every number in it is the sum of the two numbers immediately above ite.g.
Secrets of Pascal’s TriangleMany hidden patterns in it
For example:
What is the sum of squares of elements in each row?
Secrets of Pascal’s TriangleMany hidden patterns in it
What is the sum of squares of elements in each row?
Secrets of Pascal’s TriangleMany hidden patterns in it
What is the sum of squares of elements in each row?
Can we make a conjecture?
Secrets of Pascal’s TriangleMany hidden patterns in it
What is the sum of squares of elements in each row?
Can we make a conjecture?
These sum of squares are The numbers in the middle column of Triangle.
Of course, every second row contains an entry in the middle column.
So, the last value (70), sum of the squares in the 4th row is the middle element of 8th row.
Secrets of Pascal’s TriangleMany hidden patterns in it
What is the sum of squares of elements in each row?
Can we make a conjecture?
These squares are The numbers in the middle column of Triangle.
Of course, every second row contains an entry in the middle column.
So, the last value (70), sum of the squares in the 4th row is the middle element of 8th row.
Secrets of Pascal’s TriangleMany hidden patterns in it
What is the sum of squares of elements in each row?
Can we prove this conjecture?
Let’s count on both sides of the equality and show it counts the same thing..
Secrets of Pascal’s TriangleMany hidden patterns in it
What is the sum of squares of elements in each row?
Can we prove this conjecture?
Let’s count on both sides of the inequality and show it counts the same thing..
RHS: Selecting n elements from a set of 2n elements 2𝑛𝑛
Secrets of Pascal’s TriangleMany hidden patterns in it
What is the sum of squares of elements in each row?
LHS: From a set of 2𝑛 elements, make two groups of 𝑛 elements each.
Choose 𝑘 elements from 1st group and (𝑛– 𝑘 ) elements from the 2nd group. In total, we choose 𝑛 elements.
This means we choose 𝑛𝑘⋅
𝑛𝑛 − 𝑘
ways.
From n choose k 𝑛𝑘
From n choose n – k 𝑛
𝑛 − 𝑘
Secrets of Pascal’s TriangleMany hidden patterns in itWhat is the sum of squares of elements in each row?
LHS: From a set of 2𝑛 elements, make two groups of 𝑛 elements each.
Choose 𝑘 elements from 1st group and (𝑛– 𝑘 ) elements from the 2nd group. In total, we choose 𝑛 elements.
This means we choose 𝑛𝑘⋅
𝑛𝑛 − 𝑘
ways.
But 𝑛𝑘=
𝑛𝑛 − 𝑘
…. We get 𝑛𝑘
2ways.
.
From n
choose k 𝑛𝑘
From n choose n – k 𝑛
𝑛 − 𝑘
Secrets of Pascal’s TriangleMany hidden patterns in it
What is the sum of squares of elements in each row?
LHS: From a set of 2𝑛 elements, make two groups of 𝑛 elements each.
Choose 𝑘 elements from 1st group and (𝑛– 𝑘 ) elements from the 2nd group. In total, we choose 𝑛 elements.
This means we choose 𝑛𝑘⋅
𝑛𝑛 − 𝑘
ways.
But 𝑛𝑘=
𝑛𝑛 − 𝑘
…. We get 𝑛𝑘
2ways.
We have to consider all possibilities. So value of 𝑘 varies from 0 till 𝑛.
We SUM UP all cases of 𝑘:𝑛0
2+
𝑛1
2+𝑛2
2+⋯+ 𝑛
𝑛 − 1
2+ 𝑛𝑛
2
EXAMPLES – 1
Examples: --- Do it by yourself
From n choose x 𝑛𝑥
From m choose k – x 𝑚
𝑘 − 𝑥
EXAMPLES – 2
Examples: --- Do it by yourself
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Secrets of Pascal’s TriangleCan we make a conjecture?
Secrets of Pascal’s TriangleCan we make a conjecture?
Try to prove it by yourself!!!
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Secrets of Pascal’s Triangle
Watch some interesting stuff here:
TedEd Talk:
https://www.youtube.com/watch?v=XMriWTvPXHI
Numberfile documentary:
https://digg.com/video/pascals-triangle
THANK YOU
• Stay Safe and Healthy
Xie Xie!!!
Note: Class activity-4: short quiz – from the last lecture – do it till FridayAssignment-4: Will be OUT todayQuiz-3: on Weekend, related to Assignment-3
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