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Best of luck to the class of 2025 for their HSC exams. You got this! Let us know your thoughts on the HSC exams here
binomial help again sorryyyy. (1 Viewer)
- Thread starter yashbb
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Life'sHard
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Let x = 1 on LHS and RHS
wait sorry, i dont get what u do after that
ooohhhhhh ok ok tysm!!!!!You basically just sub in
^n\\=2^n\\\\\text{RHS}=\binom{n}{0}+\binom{n}{1}(1)+\binom{n}{2}(1)^2+\binom{n}{3}(1)^3\text{...}+\binom{n}{n}(1)^n\\\\=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}\text{...}+\binom{n}{n}\\\\=\sum_{k=0}^{n} \binom{n}{k}\\\\\\\therefore 2^n=\sum_{k=0}^{n} \binom{n}{k})
With these types of binomial identity questions, seeking that you find a relationship between binomial coefficients, there are essentially four approaches that you need to recognise:
} \qquad \text{Using sigma notation, state the binomial expansion of }\ (1+x)^n)
} \qquad \text{Show that }\ \sum_{k=0}^n \binom{n}{k} = 2^n)
} \qquad \text{Show that }\ \binom{n}{0} + \binom{n}{2} + \binom{n}{4} + ... = \binom{n}{1} + \binom{n}{3} + \binom{n}{3} + ...)
} \qquad \text{Find the value of }\ \sum_{k=0}^n \binom{2n}{2k})
Example Question 2
^n\ \text{as a series and hence show that }\ \sum_{k=1}^n k\binom{n}{k} = n2^{n-1})
- Substitute a value, as in the one above. This can be extended by combining results - example 1 below
- Use algebra or calculus on an expression prior to applying approach 1 - like example 2
- Form an expression by selecting terms from an identity and equating coefficients - like from the start of the 2020 MX1 HSC question
- Deriving results using combinatorics rather than, or accompanied by, algebra
Example Question 2