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  4. The value of 30 C0 30 C15 215- 30 C1 29 C14 214+ 30 C2 28 C13 213 - ldots- 30 C15 15 C0 is

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Q. The value of ${ }^{30} C_{0}{ }^{30} C_{15} 2^{15}-{ }^{30} C_{1}{ }^{29} C_{14} 2^{14}+{ }^{30} C_{2}{ }^{28} C_{13} 2^{13}$ $-\ldots-{ }^{30} C_{15}{ }^{15} C_{0}$ is

Binomial Theorem

Solution:

$S={ }^{30} C_{0}{ }^{30} C_{15} 2^{15}-{ }^{30} C_{1}{ }^{29} C_{14} 2^{14}+{ }^{30} C_{2}{ }^{28} C_{13}{ }^{213}-\ldots -{ }^{30} C_{15}{ }^{15} C_{0}$
$T_{r}={ }^{30} C_{r} \cdot{ }^{30-r} C_{15-r}(2){ }^{15-r}(-1)^{r}$
$=\frac{30 !}{(30-r) ! r !} \frac{(30-r) !}{(15-r) ! 15 !} 2^{15-r}(-1)^{r}$
$=\frac{30 !}{r !} \frac{1}{(15-r) ! 15 !} 2^{15-r}(-1)^{r}$
$=\frac{30 !}{15 ! 15 !} \frac{15 !}{r !(15-r) !} 2^{15-r}(-1)^{r}$
$={ }^{30} C_{15} \cdot{ }^{15} C_{r}(2)^{15-r}(-1)^{ r }$
$ \therefore \,\,\, S={ }^{30} C_{15}(2-1)^{15}={ }^{30} C_{15}$