A) \[^{20}{{C}_{7}}\]
B) \[^{14}{{C}_{7}}\]
C) \[^{20}{{C}_{7}}{{-}^{13}}{{C}_{7}}\]
D) \[^{20}{{C}_{7}}{{-}^{13}}{{C}_{7}}\]
Correct Answer: D
Solution :
[d] Required number of ways |
= Total number of ways without any restriction |
Number of ways in which no two consecutive soldiers are selected |
Let \[{{x}_{1}}\] be the number of soldiers before first soldier selected. |
Also, let \[{{x}_{2}},{{x}_{3}},{{x}_{4}},{{x}_{5}},{{x}_{6}}\]and \[{{x}_{7}}\] be the number of |
soldiers between \[{{1}^{st}}\] and \[{{2}^{nd}},\] \[{{2}^{nd}}\] and \[{{3}^{rd}}\], \[{{3}^{rd}}\] and \[{{4}^{th}},\] \[{{4}^{th}},\] and \[{{5}^{th}},\] \[{{5}^{th}},\] and \[{{6}^{th}},\]\[{{6}^{th}},\] and \[{{7}^{th}},\] respectively. |
And let \[{{x}_{8}}\] be the number of soldiers after the \[{{7}^{th}}\] solider selected. |
Now \[{{x}_{1}}+{{x}_{2}}+....+{{x}_{8}}=13\] |
Also \[{{x}_{1}},{{x}_{8}}\ge 0\] and \[{{x}_{2}},{{x}_{3}},....{{x}_{7}}\ge 1\] |
\[\Rightarrow \,\,\,{{x}_{1}}+{{y}_{2}}+.....+{{y}_{7}}+{{x}_{8}}=7,\] |
where \[{{y}_{2}},{{y}_{3}},............,{{y}_{7}}\ge 0\]. |
Number of solutions of above equation |
\[{{=}^{8+7-1}}{{C}_{8-1}}{{=}^{14}}{{C}_{7}}\] |
\[\therefore \] Required number of ways \[{{=}^{20}}{{C}_{7}}{{-}^{14}}{{C}_{7}}\] |
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