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Math  /  Algebra

QuestionÉvalue les expressions suivantes. Montre ton travail. (3 points) 16!4!12!16!48=\frac{16!}{4!12!} \frac{16!}{48}= b. 6×4×5!24⋅5!=6 \times 4 \times 5!24 \cdot 5!= c. 3(2×12!4!4!)3\left(\frac{2 \times 12!}{4!4!}\right) 24!20!\frac{24!}{20!} e. 6×7!5!\frac{6 \times 7!}{5!} f. 3!+4!+1!+23!+4!+1!+2 !

Studdy Solution

STEP 1

1. We are asked to evaluate each expression separately.
2. Factorials and basic arithmetic operations are involved.
3. Simplification of factorial expressions is necessary to make calculations manageable.

STEP 2

1. Simplify and evaluate expression (a).
2. Simplify and evaluate expression (b).
3. Simplify and evaluate expression (c).
4. Simplify and evaluate expression (d).
5. Simplify and evaluate expression (e).
6. Simplify and evaluate expression (f).

STEP 3

Expression (a): Simplify 16!4!12!16!48\frac{16!}{4!12!} \frac{16!}{48}.
First, simplify 16!4!12!\frac{16!}{4!12!} using the binomial coefficient formula (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}:
16!4!12!=(164)\frac{16!}{4!12!} = \binom{16}{4}
Calculate (164)\binom{16}{4}:
(164)=16×15×14×134×3×2×1=1820\binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} = 1820
Now, calculate 16!48\frac{16!}{48}:
16!48=16×15×14×13×12!48=16×15×14×13×12!48\frac{16!}{48} = \frac{16 \times 15 \times 14 \times 13 \times 12!}{48} = 16 \times 15 \times 14 \times 13 \times \frac{12!}{48}
Simplify further:
=16×15×14×13×12!48= 16 \times 15 \times 14 \times 13 \times \frac{12!}{48}
Combine both results:
1820×16!481820 \times \frac{16!}{48}
Since the expression is complex, further simplification might be needed, but this gives the general approach.

STEP 4

Expression (b): Simplify 6×4×5!×24⋅5!6 \times 4 \times 5! \times 24 \cdot 5!.
Calculate 5!5!:
5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Substitute back into the expression:
6×4×120×24×1206 \times 4 \times 120 \times 24 \times 120
Calculate the product:
=6×4×120×24×120= 6 \times 4 \times 120 \times 24 \times 120
This expression is straightforward but requires multiplication of large numbers.

STEP 5

Expression (c): Simplify 3(2×12!4!4!)3\left(\frac{2 \times 12!}{4!4!}\right).
First, simplify 12!4!4!\frac{12!}{4!4!} using the binomial coefficient formula (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}:
12!4!4!=(124)\frac{12!}{4!4!} = \binom{12}{4}
Calculate (124)\binom{12}{4}:
(124)=12×11×10×94×3×2×1=495\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495
Now, calculate the expression:
3×(2×495)=3×990=29703 \times \left(2 \times 495\right) = 3 \times 990 = 2970

STEP 6

Expression (d): Simplify 24!20!\frac{24!}{20!}.
Cancel out the common factorial terms:
24!20!=24×23×22×21\frac{24!}{20!} = 24 \times 23 \times 22 \times 21
Calculate the product:
=24×23×22×21= 24 \times 23 \times 22 \times 21

STEP 7

Expression (e): Simplify 6×7!5!\frac{6 \times 7!}{5!}.
Calculate 7!7! and 5!5!:
7!=7×6×5×4×3×2×1=50407! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 5!=1205! = 120
Substitute back into the expression:
6×5040120\frac{6 \times 5040}{120}
Simplify:
=6×42=252= 6 \times 42 = 252

STEP 8

Expression (f): Simplify 3!+4!+1!+2!3!+4!+1!+2!.
Calculate each factorial:
3!=6,4!=24,1!=1,2!=23! = 6, \quad 4! = 24, \quad 1! = 1, \quad 2! = 2
Add them together:
3!+4!+1!+2!=6+24+1+2=333! + 4! + 1! + 2! = 6 + 24 + 1 + 2 = 33

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