In multinomial expansion of (1+x5+x9)10, every term will be of form:
C(10;m,n,p)×1m×(x5)n×(x9)p,where m+n+p=10.
a) For the coefficient of x23, (5n+9p) should be equal to 23. There is only one pair exist for this condition to hold i.e (1,2).
(m,n,p) will be (7,1,2).
C(10;7,1,2)=7!1!2!10!=210(9)(8)=360The coefficient of x23 is 360.
b) For the coefficient of x32, (5n+9p) should be equal to 32. There is only one pair exist for this condition to hold i.e (1,3).
(m,n,p) will be (6,1,3).
C(10;6,1,3)=6!1!3!10!=610(9)(8)(7)=840The coefficient of x32 is 840.
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