The coefficient of x⁴ is the expansion of (1 + x + x²)¹⁰ is _____.

Solution

Maths sir · 3 steps

Pick a term from each of the 10 brackets and count every way the powers add up to 4.

1. Set up choices. Each of the 10 brackets gives 1, x or x². Say p brackets give 1, q give x and r give x². Then p + q + r = 10 and the power is q + 2r = 4.

   Number of ways = 10!/(p! q! r!)

   r (brackets giving x²)	q (brackets giving x)	p (brackets giving 1)	Ways
   0	4	6	10!/(6! 4!) = 210
   1	2	7	10!/(7! 2! 1!) = 360
   2	0	8	10!/(8! 2!) = 45

2. Why the middle row. With r = 1, q = 2 you choose 1 bracket for x² in 10 ways, then 2 of the remaining 9 for x in 36 ways. That gives 10 × 36 = 360, the biggest share.
3. So. The coefficient of x⁴ is 210 + 360 + 45 = 615.

The common slip: 255. Leaving out the mixed case with one x² and two x gives 210 + 45 = 255, not 615.

The idea to keep. For a trinomial raised to a power, list every split of the exponent with q + 2r = n and add the multinomial counts N!/(p! q! r!). Mixed cases usually carry the most weight.

Answer from the NTA final answer key. Solution by PrepWiser.