คำถาม # 832b7

ตอบ:

# = 360 * a ^ 7 * b * c ^ 2 + 840 * a ^ 6 * b ^ 3 * c + 252 * a ^ 5 * b ^ 5 #

คำอธิบาย:

#"ชื่อ"#

#p (x) = b * x + c * x ^ 2 = x (b + c * x) #

# "จากนั้นเรามี" #

# (a + p (x)) ^ 10 = sum_ {i = 0} ^ {i = 10} C (10, i) * a ^ (10-i) * p (x) ^ i #

# = sum_ {i = 0} ^ {i = 10} C (10, i) * a ^ (10-i) * x ^ i * (b + c * x) ^ i #

# "กับ" C (n, k) = (n!) / ((n-k)! k!) "(ชุดค่าผสม)" #

# = sum_ {i = 0} ^ {i = 10} C (10, i) * a ^ (10-i) * x ^ i * sum_ {j = 0} ^ {j = i} C (i, j) * b ^ (ij) * (c * x) ^ j #

# "สัมประสิทธิ์" x ^ 5 "หมายถึง" i + j = 5 => j = 5-i ""

# => C5 = sum_ {i = 0} ^ {i = 5} C (10, i) * C (i, 5-i) * a ^ (10-i) * b ^ (2 * i-5) * ค ^ (5-i) #

# => C5 = C (10,3) * C (3,2) * a ^ 7 * b * c ^ 2 + C (10,4) * C (4,1) * a ^ 6 * b ^ 3 * C + C (10,5) * C (5,0) * a ^ 5 b * ^ 5 #

# = 360 * a ^ 7 * b * c ^ 2 + 840 * a ^ 6 * b ^ 3 * c + 252 * a ^ 5 * b ^ 5 #