Question

The number of selecting at least 4 candidates from 8 candidates is
A. 270
B. 70
C. 163
D. None of these

Answer 1

Hint: In the above question they have written that total 8 candidates are there in that number of selecting the candidate at least that is\[\,r\ge 4\]. Formula which is used for this problem is \[^{n}{{c}_{r}}=\dfrac{n!}{r!(n-r)!}\] where n is the total number of candidates and r is the selecting the candidate from total number of candidates.

Complete step by step answer:
According to question there are 8 candidates out of which 4 are selected that is n =total number of candidates and r is selecting the candidates hence total number of ways is \[^{n}{{c}_{r}}\]
After substituting the value of \[\,n=8\]and \[\,r\ge 4\](at least 4 candidates are selected).
Formula which is used here is \[{{\,}^{n}}{{c}_{r}}=\dfrac{n!}{r!(n-r)!}\]
\[\therefore \]Total number of ways \[{{=}^{8}}{{c}_{4}}{{+}^{8}}{{c}_{5}}{{+}^{8}}{{c}_{6}}{{+}^{8}}{{c}_{7}}{{+}^{8}}{{c}_{8}}---(1)\]
\[^{8}{{c}_{4}}=\dfrac{8!}{4!(8-4)!}\]
After simplifying further,
\[^{8}{{c}_{4}}=\dfrac{8!}{4!(8-4)!}=70---(2)\]
Similarly we can find,
\[^{8}{{c}_{5}}=\dfrac{8!}{5!(8-5)!}=56---(3)\]
\[^{8}{{c}_{6}}=\dfrac{8!}{6!(8-6)!}=28---(4)\]
\[^{8}{{c}_{7}}=\dfrac{8!}{7!(8-7)!}=8---(5)\] \[^{8}{{c}_{8}}=\dfrac{8!}{8!(8-8)!}=1---(6)\]
After substituting the value of equation \[(2),(3),(4),(5),(6)\] in equation\[(1)\].
Total number of ways \[=\dfrac{8!}{4!(8-4)!}+\dfrac{8!}{5!(8-5)!}+\dfrac{8!}{6!(8-6)!}+\dfrac{8!}{7!(8-7)!}+\dfrac{8!}{8!(8-8)!}\]
Total number of ways \[=70+56+28+8+1\]
Total number of ways \[=163\]

So, the correct answer is “Option C”.

Note: According to given question here there are selecting at least 4 candidates which is \[\,r\ge 4\]
If the question is asked there are selecting 4 candidates then that means r is exactly 4 that is \[r=4\]
So, in this case we have to find only \[{{\,}^{n}}{{c}_{r}}\] by using the formula and substitute the value\[\,\,\,r=4\]. Total number of ways will be\[70\]. So in this way problems can be solved in a similar manner.