Show me the steps to Express as a single combination using terms from Pascal’s triangle. a) 8C4 + 8C5 b) 16C7 – 15C6 c) nCr + nCr+1
Added by Justin A.
Step 1
a) \( \binom{8}{4} + \binom{8}{5} \) ** Show more…
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(a) Complete the following table. $$\begin{array}{cccccccccc} \hline k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \left(\begin{array}{l}8 \\ k\end{array}\right) & & & & & & & & & & \\ \hline \end{array}$$ (b) Use the results in part (a) to verify that $$\left(\begin{array}{l} 8 \\ 0 \end{array}\right)+\left(\begin{array}{l} 8 \\ 1 \end{array}\right)+\left(\begin{array}{l} 8 \\ 2 \end{array}\right)+\cdots+\left(\begin{array}{l} 8 \\ 8 \end{array}\right)=2^{8}$$ (c) By taking $a=b=1$ in the expansion of $(a+b)^{n},$ show that $$\left(\begin{array}{l} n \\ 0 \end{array}\right)+\left(\begin{array}{l} n \\ 1 \end{array}\right)+\left(\begin{array}{l} n \\ 2 \end{array}\right)+\cdots+\left(\begin{array}{l} n \\ n \end{array}\right)=2^{n}$$
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Express each row of Pascal's triangle using combinations. Leave each term in the form $_{n} C_{r}$. a) $1 \quad 2 \quad 1$ b) $1 \quad 4 \quad 6 \quad 4 \quad 1$ c) $1 \quad 7 \quad 21 \quad 35 \quad 35 \quad 21 \quad 7 \quad 1$
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