| Statement I: A bag has contains 23 balls in which 7 are identical, then the number of ways of selecting 12 balls from bag is \[^{18}{{C}_{6}}{{+}^{18}}{{C}_{8}}.\] |
| Statement II: In a group has n things in which are identical, then the number of ways of selecting r things from a group is \[\sum\limits_{r=0}^{r}{^{n=p}{{C}_{r}}.}\] |
A) Statement I is true. Statement II is true; Statement B is not a correct explanation for Statement I.
B) Statement I is true. Statement II is false.
C) Statement 1 is false. Statement S is true.
D) Statement I is true, Statement H is true; Statement H is a correct explanation for Statement I.
Correct Answer: B
Solution :
Here, \[n=23,p=7,r=12(r>p)\] \[\therefore \] Required number of selections \[=\sum\limits_{r=5}^{12}{^{16}{{C}_{r}}}\] \[{{=}^{16}}{{C}_{5}}{{+}^{16}}{{C}_{6}}{{+}^{16}}{{C}_{7}}{{+}^{16}}{{C}_{8}}+\]???\[{{+}^{16}}{{C}_{12}}\] \[={{(}^{16}}{{C}_{5}}{{+}^{16}}_{6})+{{(}^{16}}{{C}_{7}}{{+}^{16}}{{C}_{8}})\] \[+{{(}^{16}}{{C}_{9}}{{+}^{16}}_{10})+{{(}^{16}}{{C}_{11}}{{+}^{16}}{{C}_{12}})\] \[={{(}^{17}}{{C}_{6}}{{+}^{17}}_{8})+{{(}^{17}}{{C}_{10}}{{+}^{17}}{{C}_{12}})\] \[={{(}^{17}}{{C}_{11}}{{+}^{17}}_{12})+{{(}^{17}}{{C}_{7}}{{+}^{17}}{{C}_{8}})\] \[{{=}^{18}}{{C}_{12}}{{+}^{18}}{{C}_{8}}\] \[{{=}^{18}}{{C}_{6}}{{+}^{18}}{{C}_{8}}\] Now, Statement II is valid only when \[r\le p\] \[\therefore \]Statement I is true and Statement II is false.
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