First slide

Multinomial expansion

Question

Consider the expansion of $(a + b + c + d)^{6}$ . Then the sum of all the coefficients of the terms

Moderate

Question

Which contains all of a, b, c and d is

A

4096
B

1560
C

3367
D

670

Solution

We have $(a + b + c + d)^{6}$
Sum of coefficients which contains all of a, b, c and d
= Number of ways of distributing six distinct objects in four boxes such that no box remains empty
$= 4^{6} -^{4} C_{1} 3^{6} +^{4} C_{2} 2^{6} -^{4} C_{1} 1^{6} = 1560$

Question

Which contains a but not b is

A

729
B

3367
C

665
D

1024

Solution

Sum of coefficients which contains a but not b
= Number of ways of distributing six distinct objects in three boxes (a, c, d)
- Number of ways of distributing six distinct objects in two boxes (c, d)
= 3⁶ - 2⁶ = 665

Question

Which contains both a and b is

A

2884
B

4032
C

1974
D

2702

Solution

Sum of coefficients which contain both a and, b
= Number of ways of distributing six distinct objects in four boxes (a, b, c, d)
- Number of ways of distributing six distinct objects in three boxes (a, c, d)
- Number of ways of distributing six distinct objects in three boxes (b, c, d)
+ Number of ways of distributing six distinct objects in two boxes (c, d)
$= 4^{6} - 2 \times 3^{6} +^{2} C_{2} \times 2^{6} = 2702$

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