Which of the following are true of false?

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,

- $(a+b) ! \neq(a !)+(b !)$

$\cdot(a \times b) ! \neq(a !) \times(b !)$