Question

1. find the value of $^6p_3$. $^6p_3 = $ enter your next step here

Explanation:

Step1: Recall permutation formula

The formula for permutations is \( _nP_r=\frac{n!}{(n - r)!} \). For \( _6P_3 \), \( n = 6 \) and \( r=3 \), so we have \( _6P_3=\frac{6!}{(6 - 3)!} \).

Step2: Simplify the denominator

Calculate \( 6-3 = 3 \), so the formula becomes \( \frac{6!}{3!} \).

Step3: Expand the factorials

We know that \( n!=n\times(n - 1)\times\cdots\times1 \), so \( 6! = 6\times5\times4\times3! \) and \( 3! = 3! \). Substituting these in, we get \( \frac{6\times5\times4\times3!}{3!} \).

Step4: Cancel out \( 3! \)

The \( 3! \) terms in the numerator and denominator cancel out, leaving \( 6\times5\times4 \).

Step5: Calculate the product

\( 6\times5 = 30 \), and \( 30\times4=120 \).

Answer:

\( 120 \)