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. $A \rightarrow 10, B \rightarrow 15120, C \rightarrow 10, D \rightarrow$ false
• B. $A \rightarrow 30, B \rightarrow 15120, C \rightarrow 20, D \rightarrow$ true
• C. A $\rightarrow 30, B \rightarrow 15120, C \rightarrow 10, D \rightarrow$ 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$
$\Rightarrow 3 !+4 ! \neq 7 !$
Hence, IV is false.
$A \rightarrow 30, B \rightarrow 15120, C \rightarrow 10, D \rightarrow \text { false }$