Question:
Which of the following are true of false?
(i) $(2+3) !=2 !+3 !$
(ii) $(2 \times 3) !=(2 !) \times(3 !)$
Solution:
Option (i) and (ii) both are false
Proofs :
For option (i)
L.H.S. $=(2+3) !=(5 !)=120$
R.H.S. $=(2 !)+(3 !)=2+6=8$
$\therefore$ L.H.S. \neqR.H.S.
For option (ii),
L.H.S. $=(2 \times 3) !=(6 !)=720$
R.H.S $=(2 !) \times(3 !)=4 \times 6=24$
$\therefore$ L.H.S. $\neq$ R.H.S.
Important Notes : for any two whole numbers a and b,
- $(a+b) ! \neq(a !)+(b !)$
$\cdot(a \times b) ! \neq(a !) \times(b !)$