Question

Fill in the blanks.
I. When $n=6,r=2$, the value of $\frac{n!}{(n-r)!}$ is ...A... .
II. When $n=9,r=5$, the value of $\frac{n!}{(n-r)!}$ is ..B... .
III. When $n=5,r=2$, the value of $\frac{n!}{r!(n-r)!}$ is ...C....
IV. $3!+4!=7!$ is ...D...
Here, A, B, C and D refer to

• A. $A\to 10,B\to 15120,C\to 10,D\to$ false
• B. $A\to 30,B\to 15120,C\to 20,D\to$ true
• C. A $\to 30,B\to 15120,C\to 10,D\to$ false
• D. None of the above

Answer 1

Answer: (C)

I. $n=6,r=2$
$\frac{n!}{(n-r)!}=\frac{6!}{(6-2)!}=\frac{6!}{4!}$
$=\frac{6\times 5\times 4!}{4!}=30$
I.. $n=9,r=5$
$\frac{n!}{(n-r)!}=\frac{9!}{4!}$
$=\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!}$
$=72\times 210$
$=15120$
III. $n=5,r=2$
$\frac{n!}{r!(n-r)!}=\frac{5!}{2!(5-2)!}=\frac{5!}{2!3!}$
$=\frac{5\times 4\times 3!}{2!3!}=\frac{5\times 4}{2!}=10$
IV. $3!+4!=6+24=30$
$7!=7\times 6\times 5\times 4\times 3\times 2\times 1$
$=42\times 120$
$=5040$
$\Rightarrow 3!+4!\ne 7!$
Hence, IV is false.
$A\to 30,B\to 15120,C\to 10,D\to \text{ false }$