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03A - Counting Notes - Great The
Typology: Study notes
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(variation of)
ൌ ʹ Ͷ ൌ 9
ͷ ൈ
ͳ
ൌ ͷ ൈ ൌ ͵Ͳ
ͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͳͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͳͲ ͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͳͳ
Bit representation^
ۀ ݊ڿ
ڿ^
^
ۀ
ۀ ݊ڿ
The binomial formula: A special case
ͳ ݔ
ൌ ͳ ͵ ݔ ͵ݔ
ݔ
ͳ ݔ
ͳ ݔ
ͳ ݔ
ͳ
ͳ^
ͳ^
ͳ^
ݔ ͳ^
ͳ^
ͳ^
ͳ^
The binomial formula: A proof
ͳ ݔ
ܿൌ
ܿ
ܿ ڮ ݔ
ݔ
ܿ
݇ ݔ
ܿ
ൌ
݇
ݔ ͳ ݔ
ൌ
ݔ^
The binomial formula: Examples
ͳ ͳ ͳݔͳ ʹ ݔ ͳݔ
ଶ
ͳ ݔ
ൌ
ͳ ݔ
ଵ
ൌ
ͳ ݔ
ଶ
ൌ
ͳ ݔ
ଷ
ൌ
ͳ ͵ ݔ ͵ݔ
ଶ
ݔͳ
ଷ
ͳ ݔ
ൌ
ݔ^
Pascal’s triangle
Ͳ Ͳ
ͳ Ͳ
ͳ ͳ
ʹ Ͳ
ʹ ͳ
ʹ ʹ
͵ Ͳ
͵ ͳ
͵ ʹ
͵ ͵
Pascal’s triangle
Pascal’s triangle
ͳ ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͷ^
ͳͲ
ͳͲ
ͷ^
ͳ
ͳ^
ͳ
ͳͷ
ͳͷ
ʹͲ
Ͳ ͳ ʹ ͵ Ͷ ͷ
ͳ
ͷ
ͳͲ
ൌ ͳͲ
݊
݇
ͳ
ͳ
Pascal’s triangle
ͳ ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͷ^
ͳͲ
ͳͲ
ͷ^
ͳ
ͳ^
ͳ
ͳͷ
ͳͷ
ʹͲ
Ͳ ͳ ʹ ͵ Ͷ ͷ
ͳ
ͷ
ൌ
ൌ
ͳ
Pascal’s triangle
ͳ ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͷ^
ͳͲ
ͳͲ
ͷ^
ͳ
ͳ^
ͳ
ͳͷ
ͳͷ
ʹͲ
Ͳ ͳ ʹ ͵ Ͷ ͷ
ͳ
ͷ
σ
ൌ ʹ
σ
ൌ ʹ
݇
݇
ͳ
ͳ^
ͳ
Pascal’s triangle
ͳ ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͷ^
ͳͲ
ͳͲ
ͷ^
ͳ
ͳ^
ͳ
ͳͷ
ͳͷ
ʹͲ
Ͳ ͳ ʹ ͵ Ͷ ͷ
ͳ
ͷ
ͳ^
ͳ
σ
ൌ
݊ൌ ͵
Pascal’s triangle
ͳ ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͳ
ͳ^
ͷ^
ͳͲ
ͳͲ
ͷ^
ͳ
ͳ^
ͳ
ͳͷ
ͳͷ
ʹͲ
Ͳ ͳ ʹ ͵ Ͷ ͷ
ͳ
ͷ
σ
ൌǫ ାଵ
݇ൌ ʹǡ ݊ൌ ͷ
ͳͲ
ͳ
Proofs from “The Book”
