here is the multinomial theorem solution...

 

P-2 R-2 O-3 I,T,N-1

 

so we have to find the coefficient of x⁴

 

x⁴ in 4! (1+Px+P²x²/2!)(1+Rx+R²x²/2!)(1+Ox+O²x²/2!+O³x³/3!)(1+Ix)(1+Tx)(1+Nx)

replacing p,r,o,i,t,n by 1

 

x⁴ in 4! (1+x+x²/2!)²(1+x+x²/2!+x³/3!)(1+x)³

x⁴ in 4! (1+2x+2x²+x³+x⁴/4)(1+x+x²/2+x³/6)(1+3x+3x²+x³)

 

opening the brackets and neglecting powers higher than x⁴

x⁴ in 4! (1+2x+2x²+x³+x⁴/4)(1+4x+(13/2)x²+(17/3)x³+3x⁴)

now gather all x⁴ terms

 

x⁴ in 4! (3x⁴+(34/3)x⁴+13x⁴+4x⁴+(1/4)x⁴)

 

so coefficient of x⁴ is 4!x(379/12)=758