Question

If $A = \{1, 3, 5, 7\}$ and $B = \{1, 2, 3, 4, 5, 6, 7, 8\}$, then the number of one-to-one functions from $A$ into $B$ is:}

• A. \( 1340 \)
• B. \( 1860 \)
• C. \( 1430 \)
• D. \( 1880 \)
• E. \( 1680 \)

Hint:

For 1-1 functions, think of it as "picking" a partner for each element of A from B without replacement. The first element of A has 8 choices, the second has 7, the third has 6, and the fourth has 5.

Answer 1

Answer: (E)

Concept: The number of one-to-one (injective) functions from a set $A$ with $n$ elements to a set $B$ with $m$ elements is given by the number of permutations of $m$ items taken $n$ at a time: \[ P(m, n) = \frac{m!}{(m-n)!} \]

Step 1: Identify the number of elements in each set.
- Set $A = \{1, 3, 5, 7\} \Rightarrow n = 4$. - Set $B = \{1, 2, 3, 4, 5, 6, 7, 8\} \Rightarrow m = 8$.

Step 2: Calculate the number of 1-1 functions.
Using the permutation formula $P(8, 4)$: \[ P(8, 4) = 8 \times 7 \times 6 \times 5 \] Calculation: \[ 8 \times 7 = 56 \] \[ 56 \times 6 = 336 \] \[ 336 \times 5 = 1680 \]