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 \[{{=}^{20}}{{C}_{7}}{{-}^{14}}{{C}_{7}}\] |
You need to login to perform this action.
You will be redirected in
3 sec
