Alternative transcript:

= 480 arrangements TIP: Objects that musn't be seperated. are treated as a single object when arranged with others. WORKE EXAMPLE 2. Find how many ways two mangoes (M) & 3 watermelons (W) can be placed in a cine it the 5 fruits are distinguishable STO IN and the mangoes: a) must not be seperated # a) M₁ M₂ W₁ W2 W3 l object [44] 120-48= 72 ways fab the 2 mangoes can be placed next to each other in ² P₂ ways. This pair is now considerd as a single object to be arranged with the 3 watermelons, giving a total of 4 objects to arrange 6) must be seperated ²P₂ x "Py = 48 ways . with no restrictions, the 5 items can be arranged in 5Ps and we know that the mangoes aren't seperated in 48 of these 1X2 x 1 x 1 282 1x ²P₂ x1 = 2 numbers. 1x 2 x 1 x 2 2P₁₂ as shown = 5! = 120 ways, WORKED EXAMPLE 3. How many odd 4-digit numesers greater than 3000 can be made from the digits 1,2,3 & 4, earn used once? Restrictions affect the digits in the thousands column and in the the whits colling. The digit at the far left Ci.e thousands commin) can be ony 3 or 4 , and the digit at the far right Cl.e units columns) can be only 1 or 3. The 3 can be placed in either of the restricted positions, so we can investigate seperately the k-digit numbers that start with 3 and the 4-digit numbers that start with 4 start with 3 we must place 1 at the far right Cone choice ), and the remaining two positions can be filled by the other 2 digits in "P₂ ways, as shown. Start with 4. We can place for 3 at the far right (two choices), and the remaining to positions can be filled by the other two digits in ²P₂ way 5.